Define $f:\mathbb{N}\to \mathbb{N}$ by $f(n)=n/2$ if $n$ is...
Define $f:\mathbb{N}\to \mathbb{N}$ by $f(n)=n/2$ if $n$ is even and $f(n)=\frac{3n+1}{2}$ if $n$ is odd.Given any integer $m\geq 1$ does there exist...
Exploring unsolved problems from Paul Erdos, one of history's most prolific mathematicians
687 open problems
Define $f:\mathbb{N}\to \mathbb{N}$ by $f(n)=n/2$ if $n$ is even and $f(n)=\frac{3n+1}{2}$ if $n$ is odd.Given any integer $m\geq 1$ does there exist...
Let $C>0$. There exists $\epsilon>0$ such that if $n$ is sufficiently large the following holds.For any $x_1,\ldots,x_n\in [-1,1]$ there exist...
For $x_1,\ldots,x_n\in [-1,1]$ let\[l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},\]which are such that $l_k(x_k)=1$ and...
For $x_1,\ldots,x_n\in [-1,1]$ let\[l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},\]which are such that $l_k(x_k)=1$ and...
For $x_1,\ldots,x_n\in [-1,1]$ let\[l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},\]which are such that $l_k(x_k)=1$ and...
For $x_1,\ldots,x_n\in [-1,1]$ let\[l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},\]which are such that $l_k(x_k)=1$ and...
Let $B_1$ be the Boolean algebra of sets of integers modulo sets of density $0$ (that is, in which two sets are equivalent if and only if they differ...
Let $f:\mathbb{N}\to \mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b)=1$). Let\[A=\{ n \geq 1: f(n+1)< f(n)\}.\]If $\lvert...
Let $f\in \mathbb{C}[z]$ be a monic polynomial of degree $n$, all of whose roots satisfy $\lvert z\rvert\leq 1$. Let\[E= \{ z : \lvert f(z)\rvert...
Let $f(z)$ be an entire function which is not a monomial. Let $\nu(r)$ count the number of $z$ with $\lvert z\rvert=r$ such that $\lvert...
A positive odd integer $m$ such that none of $2^km+1$ are prime for $k\geq 0$ is called aSierpinski number. We say that a set of primes $P$ is a...
Let $1\leq d_1
If $G$ is a finite graph and $A,B$ are disjoint sets of vertices then we call $A,B$ anticomplete if there are no edges between $A$ and $B$.If...
Let $p>q\geq 2$ be two coprime integers. We call $n$ representable if it is the sum of integers of the form $p^kq^l$, none of which divide each...
Let $f(N)$ be the size of the largest subset $A\subseteq \{1,\ldots,N\}$ such that every $n\in A+A$ is squarefree. Estimate $f(N)$. In particular, is...
Let\[A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\textrm{ finite}\right\}.\]If $k\geq 2$, then does $A$ contain only finitely many $k$th powers?...
Let $r\geq 2$. A number $n$ is $r$-powerful if for every prime $p$ which divides $n$ we have $p^r\mid n$. Is every large integer the sum of at most...
Let $p(n)$ denote thepartition functionof $n$ and let $F(n)$ count the number of distinct prime factors of\[\prod_{1\leq k\leq n}p(k).\]Does $F(n)\to...
The anti-Ramsey number $\mathrm{AR}(n,G)$ is the maximum possible number of colours in which the edges of $K_n$ can be coloured without creating a...
Let $f(n)$ be the maximum possible chromatic number of a triangle-free graph on $n$ vertices. Estimate $f(n)$.
Let $A$ be an infinite sequence of integers such that every $n\in A+A$ is squarefree. How fast must $A$ grow?
If $u=\{u_1
If $1=d_1<\cdots
Let $A$ be a set of $n$ integers. How many distinct $d$ can occur as the common difference of a three-term arithmetic progression in $A$? Are there...
Let $1
Let $g(k)>k+1$ be the smallest $n$ such that all prime factors of $\binom{n}{k}$ are $>k$. Estimate $g(k)$.
For all $n\geq 2k$ the least prime factor of $\binom{n}{k}$ is $\leq \max(n/k,k)$, with only finitely many exceptions.
For $n\geq 2k$ we define the deficiency of $\binom{n}{k}$ as follows. If $\binom{n}{k}$ is divisible by a prime $p\leq k$ then the deficiency is...
Let $f_r(n)$ be maximal such that, if a graph $G$ has the property that every subgraph $H$ on $m$ vertices is the union of a graph with chromatic...
Let $G$ be a $K_4$-free graph with chromatic number $4$. Must $G$ contain an odd cycle with at least two diagonals?More generally, is there some...
Let $g_d(n)$ be minimal such that every collection of $g_d(n)$ points in $\mathbb{R}^d$ determines at least $n$ many distinct distances. Estimate...
Let $f_d(n)$ be the minimal $m$ such that any set of $m$ points in $\mathbb{R}^d$ contains a set of $n$ points such that any two determined distances...
Let $f(n)$ be minimal such that every set of $n$ points in $\mathbb{R}^2$ contains at most $f(n)$ many sets of four points which are 'degenerate' in...
Let $g(n)$ be minimal such that any set of $n$ points in $\mathbb{R}^2$ contains the vertices of at most $g(n)$ many triangles with the same area....
Let $f_d(n)$ be minimal such that, in any set of $n$ points in $\mathbb{R}^d$, there exist at most $f_d(n)$ pairs of points which distance $1$ apart....
Let $f_d(n)$ be minimal such that in any collection of $n$ points in $\mathbb{R}^d$, all of distance at least $1$ apart, there are at most $f_d(n)$...
Let $d\geq 3$, and let $f_d(n)$ be the minimal $m$ such that every set of $n$ points in $\mathbb{R}^d$ determines at least $m$ distinct distances....
Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no three on a line. Does $A$ determine at least $\lfloor n/2\rfloor$ distinct distances? In...
Let $r\geq 3$. There exists $c_r>r^{-r}$ such that, for any $\epsilon>0$, if $n$ is sufficiently large, the following holds.Any $r$-uniform...
Let $S$ be the set of all $m\geq 1$ such that there exists a prime $p\not\equiv 1\pmod{m}$ such that $m!+1\equiv 0\pmod{p}$. Does\[\lim \frac{\lvert...
Let $A(x)$ count the number of composite $u
For any prime $p$, let $f(p)$ be the least integer such that $f(p)!+1\equiv 0\pmod{p}$.Is it true that there are infinitely many $p$ for which...
Is there a finite set of unit line segments (rotated and translated copies of $(0,1)$) in the unit square, no two of which intersect, which are...
Let $f(n)$ be maximal such that, given any $n$ points in $\mathbb{R}^2$, there exist $f(n)$ points such that no two are distance $1$ apart. Estimate...
Does every graph with chromatic number $\aleph_1$ contain a countable subgraph which is infinitely vertex-connected?
Let $G$ be a graph given by $n$ points in $\mathbb{R}^2$, where any two distinct points are at least distance $1$ apart, and we draw an edge between...
Are there infinitely many primes $p$ such that $p=2^kq+1$ for some prime $q$ and $k\geq 0$? Or $p=2^k3^lq+1$?
Let $k\geq 2$ and define $n_k\geq 2k$ to be the least value of $n$ such that $n-i$ divides $\binom{n}{k}$ for all but one $0\leq i
Let $f(n)$ be the size of the largest subset $A\subseteq \{1,\ldots,n\}$ such that there are no three distinct elements $a,b,c\in A$ such that $a\mid...
How many solutions are there to\[\sigma(a)+\sigma(b)=\sigma(a+b)\]with $a+b\leq x$, where $\sigma$ is the sum of divisors function? Is it $\sim cx$...
Let $f(n)$ count the number of solutions to $k\sigma(k)=n$, where $\sigma(k)$ is the sum of divisors of $k$. Is it true that $f(n)\leq...
Are there infinitely many primes $p$ such that $p-k!$ is composite for each $k$ such that $1\leq k!
Let $C(x)$ count the number ofCarmichael numbersin the interval $[1,x]$. Is it true that $C(x)=x^{1-o(1)}$?
Let $k\geq 2$. Does there exist a prime $p$ and consecutive intervals $I_1,\ldots,I_k$ such that\[\prod_{n\in I_i}n \equiv 1\pmod{p}\]for all $1\leq...
A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of...
Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$.Is it true that $f(n)=o(n)$? Or...
Call a number $k$-perfect if $\sigma(n)=kn$, where $\sigma(n)$ is the sum of the divisors of $n$. Must $k=o(\log\log n)$?
A unitary divisor of $n$ is $d\mid n$ such that $(d,n/d)=1$. A number $n\geq 1$ is aunitary perfect numberif it is the sum of its unitary divisors...
Is it true that if $a_1
Let $t>1$ be a rational number. Is\[\sum_{n=1}^\infty\frac{1}{t^n-1}=\sum_{n=1}^\infty \frac{\tau(n)}{t^n}\]irrational, where $\tau(n)$ counts the...
Let $z_1,\ldots,z_n\in \mathbb{C}$ with $\lvert z_i-z_j\rvert\leq 2$ for all $i,j$, and\[\Delta(z_1,\ldots,z_n)=\prod_{i\neq j}\lvert...
Let $f(z)=\prod_{i=1}^n(z-z_i)\in\mathbb{C}[x]$ where $\lvert z_i\rvert\leq 1$ for all $i$. If $\Lambda(f)$ is the maximum of the lengths of the...
Let $f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z]$ with $\lvert z_i\rvert < 1$ for all $i$.Must there always exist a path of length less than $2$...
Let $F\subseteq \mathbb{C}$ be a closed infinite set, and let $\mu(F)$ be the infimum of\[\lvert \{ z: \lvert f(z)\rvert < 1\}\rvert,\]as $f$ ranges...
Let $f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z]$ with $\lvert z_i\rvert \leq 1$ for all $i$. Let $\rho(f)$ be the radius of the largest disc which is...
Determine the infimum and supremum of\[\lvert \{ x\in \mathbb{R} : \lvert f(x)\rvert < 1\}\rvert\]as $f\in \mathbb{R}[x]$ ranges over all...
Is there a constant $c>0$ such that every graph on $2^n$ vertices with minimum degree $>(1-c)2^n$ contains the $n$-dimensional hypercube $Q_n$?
Let $h(n)$ be such that every graph on $n$ vertices with $>n^2/4$ many edges contains a triangle whose vertices have degrees summing to at least...
We say that a graph is $4$-chromatic critical if it has chromatic number $4$, and removing any edge decreases the chromatic number to $3$.Is there,...
If $R(k,l)$ is the Ramsey number then prove the existence of some $c>0$ such that\[\lim_k \frac{R(k+1,k)}{R(k,k)}> 1+c.\]
If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$,...
Is there a constant $c_t$, where $c_t\to \infty$ as $t\to \infty$, such that if $\mathcal{F}$ is a finite family of finite sets, all of size at least...
Is it true that, for every $k\geq 3$, there is a constant $c_k>0$ such that\[\mathrm{ex}(n,G_k) \ll n^{3/2-c_k},\]where $G_k$ is the bipartite graph...
Let $f(n;r,k)$ be the maximal number of edges in an $r$-uniform hypergraph which contains no set of $k$ many independent edges.For all $r\geq...
Let $f(n,k)$ be such that every graph on $n$ vertices and $k$ edges can be partitioned into at most $f(n,k)$ edge-disjoint complete graphs. Estimate...
Let $h(n)$ be minimal such that there is a graph on $n$ vertices with $n+h(n)$ edges which contains a cycle on $k$ vertices, for all $3\leq k\leq n$....
Let $R(k,l)$ be the Ramsey number, so the minimal $n$ such that every graph on at least $n$ vertices contains either a $K_k$ or an independent set on...
Let $h_3(k)$ be the minimal $n$ such that there exists a triangle-free graph on $n$ vertices with chromatic number $k$. Find an asymptotic for...
Let $f_r(n)$ be minimal such that every graph on $n$ vertices with $\geq f_r(n)$ edges and chromatic number $\geq r$ contains a triangle. Determine...
Let $\frac{a_1}{b_1},\frac{a_2}{b_2},\ldots$ be theFarey fractionsof order $n\geq 4$. Let $f(n)$ be the largest integer such that if $1\leq k
Let $c>0$. If $x$ is sufficiently large then does there exist $n\leq x$ such that the values of $\phi(n+k)$ are all distinct for $1\leq k\leq (\log...
Are there infinitely many solutions to $\phi(n)=\phi(n+1)$, where $\phi$ is the Euler totient function?
For any $0<\alpha<1$, let\[f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}-\{ \alpha k\}).\]Does $f(\alpha,n)$ have an asymptotic...
Call $x_1,x_2,\ldots \in (0,1)$ well-distributed if, for every $\epsilon>0$, if $k$ is sufficiently large then, for all $n>0$ and intervals...
Let $n_1
Let $n_1
The independent set sequence of any tree or forest is unimodal.In other words, if $i_k(G)$ counts the number of independent sets of vertices of size...
Let $x_1
Let $f=a_0+\cdots+a_dx^d\in \mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\ldots,z_d$ with corresponding arguments...
Let $x_1,x_2,\ldots \in (0,1)$ be an infinite sequence and let\[A_k=\limsup_{n\to \infty}\left\lvert \sum_{j\leq n} e(kx_j)\right\rvert,\]where...
For any fixed $k\geq 3$,\[R(k,n) \gg \frac{n^{k-1}}{(\log n)^c}\]for some constant $c=c(k)>0$.
Is it true that, for every prime $p$, there is a prime $q
Let $n\geq 2$ and $\pi(n)
If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then some vertex has at least $\lfloor \frac{n}{2}\rfloor$ different distances to...
Let $k\geq 2$, and let $f_k(n)$ count the number of solutions to\[n=p_1^k+\cdots+p_k^k,\]where the $p_i$ are prime numbers. Is it true that $\limsup...
Let $f\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k\neq 2^l$ for any $l\geq 1$) such that the leading...
Let $f\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $d\geq 2$. Let $F_f(n)$ be maximal such that there exists $1\leq m\leq n$ with $f(m)$...
Let $f\in \mathbb{Z}[x]$ be an irreducible non-constant polynomial such that $f(n)\geq 1$ for all large $n\in\mathbb{N}$. Does there exist a constant...
Does there exist a constant $C>1$ such that, for every $n\geq 2$, there exists a sequence $z_i\in \mathbb{C}$ with $z_1=1$ and $\lvert z_i\rvert \geq...
Let $\alpha>1$ be irrational. Are there infinitely many primes $p$ such that $\lfloor p\alpha\rfloor$ is also prime?
Let $p(a,d)$ be the least prime congruent to $a\pmod{d}$. Does there exist a constant $c>0$ such that, for all large $d$,\[p(a,d) > (1+c)\phi(d)\log...
Let $h(k)$ be Jacobsthal's function, defined to as the minimal $m$ such that, if $n$ has at most $k$ prime factors, then in any set of $m$...
Let $Q(x)$ count the number of squarefree integers in $[1,x]$. Determine the order of magnitude in the error term in the...
Let $u_n=p_n/n$, where $p_n$ is the $n$th prime. Does the set of $n$ such that $u_n
Let $f(n)$ be the maximal $k$ such that in any set $A\subset \mathbb{R}$ of size $n$ there is a subset $B\subseteq A$ of size $\lvert B\rvert\geq k$...
Let $k(n)$ be the maximal $k$ such that there exists $m\leq n$ such that each of the integers\[m+1,\ldots,m+k\]are divisible by at least one prime...
Let $f(k)$ be the minimal $n$ such that every set of $n$ consecutive integers $>k$ contains an integer divisible by a prime $>k$. Estimate $f(k)$.
Let $r,k\geq 2$ be fixed. Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no $k$ points on a line. Determine the threshold $f_{r,k}(n)$ such...
Let $A\subset \mathbb{R}^2$ be a set of size $n$ and let $\{d_1,\ldots,d_k\}$ be the set of distinct distances determined by $A$. Let $f(d)$ be the...
If $C,D\subseteq \mathbb{R}^2$ then the distance between $C$ and $D$ is defined by\[\delta(C,D)=\inf_{\substack{c\in C\\ d\in D}}\| c-d\|.\]Let...
Let\[s(n)=\sigma(n)-n=\sum_{\substack{d\mid n\\ d
Let $1=a_1
Let $A\subset \{ x\in \mathbb{R}^2 : \lvert x\rvert
Is there an infinite sequence of distinct Gaussian primes $x_1,x_2,\ldots$ such that\[\lvert x_{n+1}-x_n\rvert \ll 1?\]
Let $1
Let\[f(n) = \sum_{p
Let $S\subset \mathbb{R}$ be a set containing no solutions to $a+b=c$. Must there be a set $A\subseteq \mathbb{R}\backslash S$ of cardinality...
Is there a function $f(n)$ and a $k$ such that in any $k$-colouring of the integers there exists a sequence $a_1<\cdots$ such that $a_n
Let $F(x)$ be the maximal $k$ such that there exist $n+1,\ldots,n+k\leq x$ with $\tau(n+1),\ldots,\tau(n+k)$ all distinct (where $\tau(m)$ counts the...
A critical vertex, edge, or set of edges, is one whose deletion lowers the chromatic number.Let $k\geq 4$ and $r\geq 1$. Must there exist a graph $G$...
Let $A$ be the set of powerful numbers (if $p\mid n$ then $p^2\mid n$). Is it true that\[1_A\ast 1_A(n)=n^{o(1)}\]for every $n$?
Let $h(n)$ count the number of powerful (if $p\mid m$ then $p^2\mid m$) integers in $[n^2,(n+1)^2)$. Estimate $h(n)$. In particular is there some...
Let $r\geq 3$. A number $n$ is $r$-powerful if for every prime $p$ which divides $n$ we have $p^r\mid n$.Are there infinitely many integers which...
Let $r\geq 2$. An $r$-powerful number $n$ is one such that if $p\mid n$ then $p^r\mid n$.If $r\geq 4$ then can the sum of $r-2$ coprime $r$-powerful...
Let $A=\{n_1
Are\[2^n\pm 1\]and\[n!\pm 1\]powerful (i.e. if $p\mid m$ then $p^2\mid m$) for only finitely many $n$?
For any integer $n=\prod p^{k_p}$ let $Q_2(n)$ be the powerful part of $n$, so that\[Q_2(n) = \prod_{\substack{p\\ k_p\geq 2}}p^{k_p}.\]Is it true...
Let $h_t(d)$ be minimal such that every graph $G$ with $h_t(d)$ edges and maximal degree $\leq d$ contains two edges whose shortest path between them...
If $n(n+1)=2^k3^lm$, where $(m,6)=1$, then is it true that\[\limsup_{n\to \infty} \frac{2^k3^l}{n\log n}=\infty?\]
Let $p_k$ denote the $k$th prime. For infinitely many $r$ there are at least two integers $p_r
Let $k_1\geq k_2\geq 3$. Are there only finitely many $n_2\geq n_1+k_1$ such that\[\prod_{1\leq i\leq k_1}(n_1+i)\textrm{ and }\prod_{1\leq j\leq...
Is it true that, for every $r$, there is a $k$ such that if $I_1,\ldots,I_r$ are disjoint intervals of consecutive integers, all of length at least...
Let $k\geq 2$ be large and let $S(k)$ be the minimal $x$ such that there is a positive density set of $n$ where\[n+1,n+2,\ldots,n+k\]are all...
Let $\alpha,\beta\in (0,1)$ and let $P(n)$ denote the largest prime divisor of $n$. Does the density of integers $n$ such that $P(n)
Let $g_k(n)$ denote the largest possible chromatic number of a graph with $n$ vertices which contains no $K_k$.Is it true that, for $k\geq...
Is there a graph $G$ with vertex set $\omega_2^2$ and chromatic number $\aleph_2$ such that every subgraph whose vertices have a lesser type has...
Is there a graph with $\aleph_2$ vertices and chromatic number $\aleph_2$ such that every subgraph on $\aleph_1$ vertices has chromatic number...
Let $k\geq 4$ and $f_k(n)$ be the largest number of edges in a graph on $n$ vertices which has chromatic number $k$ and is critical (i.e. deleting...
Are there infinitely many $n$ such that if\[n(n+1) = \prod_i p_i^{k_i}\]is the factorisation into distinct primes then all exponents $k_i$ are...
If\[n! = \prod_i p_i^{k_i}\]is the factorisation into distinct primes then let $h(n)$ count the number of distinct exponents $k_i$.Prove that there...
Let $\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges that is Ramsey for...
Is there an entire non-zero function $f:\mathbb{C}\to \mathbb{C}$ such that, for any infinite sequence $n_1
Let $f(n)$ be minimal such that there is a tournament (a complete directed graph) on $f(n)$ vertices such that every set of $n$ vertices is dominated...
Let $m(n)$ be minimal such that there is an $n$-uniform hypergraph with $m(n)$ edges which is $3$-chromatic. Estimate $m(n)$.
Estimate the maximum of $F(A,B)$ as $A,B$ range over all subsets of $\{1,\ldots,N\}$, where $F(A,B)$ counts the number of $m$ such that $m=ab$ has...
If $\tau(n)$ counts the divisors of $n$ then let\[f(n)=\sum_{1\leq k\leq n}\tau(2^k-1).\]Does $f(2n)/f(n)$ tend to a limit?
Is there a necessary and sufficient condition for a sequence of integers $b_1
Let $2=p_1
If $\omega(n)$ counts the number of distinct prime factors of $n$, then is it true that, for every $k\geq 1$,\[\liminf_{n\to \infty}\sum_{0\leq...
For $k\geq 0$ and $n\geq 1$ let $v(n,k)$ count the prime factors of $n+k$ which do not divide $n+i$ for $0\leq i
What is the size of the largest $A\subseteq \{1,\ldots,n\}$ such that if $a\leq b\leq c\leq d\in A$ are such that $abcd$ is a square then $ad=bc$?
Is there an absolute constant $K$ such that, for every $C>0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{1/2},n^{1/2}+C...
Let $\epsilon>0$. Is it true that, for all large $n$, the number of divisors of $n$ in $(n^{1/2},n^{1/2}+n^{1/2-\epsilon})$ is $O_\epsilon(1)$?
For integer $n\geq 1$ we define the factor difference set of $n$ by\[D(n) = \{\lvert a-b\rvert : n=ab\}.\]Is it true that, for every $k\geq 1$, there...
Is it true that, for any $n$, if $d_1<\cdots
For $A\subseteq \{1,\ldots,n\}$ let $G(A)$ be the graph with vertex set $A$, where two integers are joined by an edge if they are coprime.Is it true...
Let $A\subset\mathbb{N}$ be an additive basis of order $k$ which is minimal, in the sense that if $B\subset A$ is any infinite set then $A\backslash...
Call a set $S\subseteq \{1,\ldots,n\}$ admissible if $(a,b)=1$ for all $a\neq b\in S$. Let\[G(n) = \max_{S\subseteq \{1,\ldots,n\}} \sum_{a\in...
If $n=\prod_{1\leq i\leq t} p_i^{k_i}$ is the factorisation of $n$ into distinct primes then let\[f(n)=\sum p_i^{\ell_i},\]where $\ell_i$ is chosen...
Let $A=\{a_1
Let $A=\{a_1
Let $A=\{a_1
Consider the two-player game in which players alternately choose integers from $\{2,3,\ldots,n\}$ to be included in some set $A$ (the same set for...
Let $k\geq 3$ and $A$ be an additive basis of order $k$. Does there exist a constant $c=c(k)>0$ such that if $r(n)\geq c\log n$ for all large $n$...
If $A_1,A_2$ are disjoint additive bases of order $2$ (i.e. $A_i+A_i$ contains all large integers) then must $A=A_1\cup A_2$ contain a minimal...
If $A$ is an additive basis of order $2$, and $1_A\ast 1_A(n)\to \infty$ as $n\to \infty$, then must $A$ contain a minimal additive basis of order...
Let $k\geq 3$ and $g_k(N)$ be minimal such that if $A\subseteq \{1,\ldots,2N\}$ has $\lvert A\rvert \geq N+g_k(N)$ then there exist integers...
There exists a constant $C>0$ such that, for all large $N$, if $A\subseteq \{1,\ldots,N\}$ has size at least $\frac{5}{8}N+C$ then there are distinct...
Let $A\subseteq \{1,\ldots N\}$ be a set such that there exists at most one $n$ with more than one solution to $n=a+b$ (with $a\leq b\in A$)....
Let $r\geq 2$ and let $A\subseteq \{1,\ldots,N\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a+b$ with $a\leq b$ for...
Let $h(n)$ be such that, for any $m\geq 1$, in the interval $(m,m+h(n))$ there exist distinct integers $a_i$ for $1\leq i\leq \pi(n)$ such that...
Let $t\geq 1$ and let $d_t$ be the density of the set of integers $n\in\mathbb{N}$ for which $t$ can be represented as the sum of distinct divisors...
Let $A\subseteq \{1,\ldots,N\}$ be such that there is no solution to $at=b$ with $a,b\in A$ and the smallest prime factor of $t$ is $>a$. Estimate...
Let $m=m(n,k)$ be minimal such that in any collection of sets $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ there must exist a sunflower of size $k$ -...
Let $k\geq 3$ and $f_k(N)$ be the maximum value of $\sum_{n\in A}\frac{1}{n}$, where $A$ ranges over all subsets of $\{1,\ldots,N\}$ which contain no...
If $\pi(x)$ counts the number of primes in $[1,x]$ then is it true that (for large $x$ and $y$)\[\pi(x+y) \leq \pi(x)+\pi(y)?\]
Let $n_k$ denote the $k$th primorial, i.e. the product of the first $k$ primes.If $1=a_1
Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n=t$ has no solutions for $n\leq...
Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $h(x)$ be maximal such that for some $n
Let $\epsilon>0$. Is there some $r\ll_\epsilon 1$ such that the density of integers of the form $2^k+n$, where $k\geq 0$ and $n$ has at most $r$...
Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors, $x+1,y+1$ have the same prime factors, and $x+2,y+2$...
Is it true that, for every integer $t\geq 1$, there is some integer $a$ such that\[\binom{n}{k}=a\](with $1\leq k\leq n/2$) has exactly $t$ solutions?
Is the maximum size of a set $A\subseteq \{1,\ldots,N\}$ such that $ab+1$ is never squarefree (for all $a,b\in A$) achieved by taking those $n\equiv...
Let $A\subset \mathbb{N}$ be an infinite set for which there exists some $\epsilon>0$ such that in any subset of $A$ of size $n$ there is a subset of...
Let $A\subset \mathbb{R}^2$ be an infinite set for which there exists some $\epsilon>0$ such that in any subset of $A$ of size $n$ there are always...
Let $C>0$. Is it true that the set of integers of the form\[n=b_1+\cdots+b_t\textrm{ with }b_1<\cdots
Let $A\subseteq \{1,\ldots,N\}$ be such that, for all $a,b\in A$, the product $ab$ is not squarefree.Is the maximum size of such an $A$ achieved by...
Are the squares Ramsey $2$-complete?That is, is it true that, in any 2-colouring of the square numbers, every sufficiently large $n\in \mathbb{N}$...
Let $G$ be a graph on $3n$ vertices formed by taking $n$ vertex disjoint triangles and adding a Hamiltonian cycle (with all new edges) between these...
Let $t_n$ be minimal such that $\{n+1,\ldots,n+t_n\}$ contains a subset whose product with $n$ is a square number (and let $t_n=0$ if $n$ is itself...
Let $f(N)$ be the size of the largest quasi-Sidon subset $A\subset\{1,\ldots,N\}$, where we say that $A$ is quasi-Sidon if\[\lvert...
Let $1\leq a_1
Let $f(n)$ be maximal such that any $n$ points in $\mathbb{R}^2$, with no three on a line, determine at least $f(n)$ different convex subsets....
Let $k\geq 2$ and $A_k\subseteq [0,1]$ be the set of $\alpha$ such that there exists some $\beta(\alpha)>\alpha$ with the property that, if...
Let $r\geq 2$ and $G$ be a $r$-uniform hypergraph with chromatic number $3$ (that is, there is a $3$-colouring of the vertices of $G$ such that no...
Does there exist a $k>2$ such that the $k$-sized subsets of $\{1,\ldots,2k\}$ can be coloured with $k+1$ colours such that for every $A\subset...
Does there exist a $3$-critical $3$-uniform hypergraph in which every vertex has degree $\geq 7$?
Does there exist an absolute constant $c>0$ such that, for all $r\geq 2$, in any $r$-uniform hypergraph with chromatic number $3$ there is a vertex...
Let $r\geq 3$ and $k$ be sufficiently large in terms of $r$. Is it true that every $r$-uniform hypergraph with chromatic number $k$ has at...
Let $h(n)$ be maximal such that in any $n$ points in $\mathbb{R}^2$ (with no three on a line and no four on a circle) there are at least $h(n)$ many...
We say that $a,b\in \mathbb{N}$ are anamicable pairif $\sigma(a)=\sigma(b)=a+b$. Are there infinitely many amicable pairs? If $A(x)$ counts the...
Let $A\subset\mathbb{N}$ be the set of cubes. Is it true that\[1_A\ast 1_A(n) \ll (\log n)^{O(1)}?\]
Is it true that, for any $a\in\mathbb{Z}$, there are infinitely many $n$ such that\[\phi(n) \mid n+a?\]
Let $n_k$ be minimal such that if $n_k$ points in $\mathbb{R}^2$ are in general position then there exists a subset of $k$ points such that all...
Are there infinitely many $n$ such that, for all $k\geq 1$,\[\tau(n+k)\ll k?\]
Is there an absolute constant $C>0$ such that every integer $n$ with $\sigma(n)>Cn$ is the distinct sum of proper divisors of $n$?
Let $h(x)$ count the number of integers $1\leq a
Let $\alpha\geq 1$. Is there a sequence of integers $n_k,m_k$ such that $n_k/m_k\to \alpha$ and $\sigma(n_k)=\sigma(m_k)$ for all $k\geq 1$, where...
Does the set of integers of the form $n+\phi(n)$ have positive (lower) density?
Let $g(n)$ count the number of $m$ such that $\phi(m)=n$. Is it true that, for every $\epsilon>0$, there exist infinitely many $n$ such that\[g(n) >...
Let $H(n)$ be the smallest integer $l$ such that there exist $k
Let $f(N)$ be maximal such that there exists $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert=\lfloor N^{1/2}\rfloor$ such that $\lvert (A+A)\cap...
Let $A$ be a finite set of integers such that $\lvert A+A\rvert \ll \lvert A\rvert$. Is it true that\[\lvert AA\rvert \gg \frac{\lvert...
Let $k\geq 3$ and define $g_k(n)$ to be the minimal $N$ such that $\{1,\ldots,N\}$ contains some $A$ of size $\lvert A\rvert=n$ such that\[\langle...
Let $G$ be a graph with $2n+1$ vertices and $n^2+n+1$ edges. Must $G$ contain two vertices of the same degree which are joined by a path of length...
Let $k\geq 3$ and $n$ be sufficiently large. Is it true that if $G$ is a graph with $n$ vertices and $2n-2$ edges such that every proper induced...
Let $k\geq 2$ and $G$ be a graph with $n\geq k-1$ vertices and\[(k-1)(n-k+2)+\binom{k-2}{2}+1\]edges. Does there exist some $c_k>0$ such that $G$...
Let $h(n)$ be minimal such that every graph on $n$ vertices where every set of $7$ vertices contains a triangle (a copy of $K_3$) must contain a...
Is it true that\[\frac{R(n+1)}{R(n)}\geq 1+c\]for some constant $c>0$, for all large $n$? Is it true that\[R(n+1)-R(n) \gg n^2?\]
Suppose $n\equiv 1\pmod{m}$. We say that an edge-colouring of $K_n$ using $m$ colours is balanced if every vertex sees exactly $\lfloor n/m\rfloor$...
Does there exist some $\epsilon>0$ such that, for all sufficiently large $n$, there exists a graph $G$ on $n$ vertices with at least $\epsilon n^2$...
Let $k\geq 3$ and define $F_k(n)$ to be the minimal $r$ such that there is a graph $G$ on $n$ vertices with $\lfloor n^2/4\rfloor+1$ many edges such...
Let $c,\epsilon>0$ and $n$ be sufficiently large. If $A\subset \mathbb{N}$ has $\lvert A\rvert=n$ and $G$ is any graph on $A$ with at least $n^{1+c}$...
The bipartition number $\tau(G)$ of a graph $G$ is the smallest number of pairwise edge disjoint complete bipartite graphs whose union is $G$. The...
Let $A\subseteq \{1,\ldots,n\}$ with $\lvert A\rvert \leq n^{1/2}$. Must there exist some $B\subset\mathbb{Z}$ with $\lvert B\rvert=o(n^{1/2})$ such...
For which functions $g(n)$ with $n>g(n)\geq (\log n)^2$ is there a graph on $n$ vertices in which every induced subgraph on $g(n)$ vertices contains...
Let $f(m,n)$ be maximal such that any graph on $n$ vertices in which every induced subgraph on $m$ vertices has an independent set of size at least...
We call a graph $H$ $D$-balanced (or $D$-almost-regular) if the maximum degree of $H$ is at most $D$ times the minimum degree of $H$.Is it true that...
Is it true that any $K_r$-free graph on $n$ vertices with average degree $t$ contains an independent set on\[\gg_r \frac{\log t}{t}n\]many vertices?
If $G$ is a graph on $n$ vertices containing no independent set on $>n^{1/2}$ vertices then there is a set of $\leq n^{1/2}$ vertices containing $\gg...
If $G$ is a graph on $n$ vertices which has no two adjacent vertices of degree $\geq 3$ then\[R(G)\ll n,\]where the implied constant is absolute.
The list chromatic number $\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$...
Let $t(n)$ be the minimum number of points in $\{1,\ldots,n\}^2$ such that the $\binom{t}{2}$ lines determined by these points cover all points in...
Let $f(d)$ be the maximal acyclic chromatic number of any graph with maximum degree $d$ - that is, the vertices of any graph with maximum degree $d$...
Let $k\geq 2$ and let $g_k(n)$ be the largest possible size of $A\subseteq \{1,\ldots,n\}$ such that every $m$ has $
Let $g(n)$ be the maximal size of $A\subseteq \{1,\ldots,n\}$ such that the products $\prod_{n\in S}n$ are distinct for all $S\subseteq A$. Is it...
Is it true that every $3$-uniform hypergraph on $3n$ vertices with at least $n^3+1$ edges must contain either a subgraph on $4$ vertices with $3$...
Let $F(n)$ be the maximum possible size of a subset $A\subseteq\{1,\ldots,n\}$ such that $a\nmid bc$ whenever $a,b,c\in A$ with $a\neq b$ and $a\neq...
Let $f(n)$ be maximal such that in any $A\subset \mathbb{Z}$ with $\lvert A\rvert=n$ there exists some sum-free subset $B\subseteq A$ with $\lvert...
Let $g(n)$ be minimal such that there exists $A\subseteq \{0,\ldots,n\}$ of size $g(n)$ with $\{0,\ldots,n\}\subseteq A+A$. Estimate $g(n)$. In...
Let $l(n)$ be maximal such that if $A\subset\mathbb{Z}$ with $\lvert A\rvert=n$ then there exists a sum-free $B\subseteq A$ with $\lvert B\rvert \geq...
Let $h(n)$ be maximal such that if $A\subseteq \mathbb{Z}$ with $\lvert A\rvert=n$ then there is $B\subseteq A$ with $\lvert B\rvert \geq h(n)$ such...
Let $f(n)$ be maximal such that if $B\subset (2n,4n)\cap \mathbb{N}$ there exists some $C\subset (n,2n)\cap \mathbb{N}$ such that $c_1+c_2\not\in B$...
Let $g(n)$ be maximal such that given any set $A\subset \mathbb{R}$ with $\lvert A\rvert=n$ there exists some $B\subseteq A$ of size $\lvert...
Let $\epsilon>0$. Is there some set $A\subset \mathbb{N}$ of density $>1-\epsilon$ such that $a_1\cdots a_r=b_1\cdots b_s$ with $a_i,b_j\in A$ can...
Let $A,B\subseteq \mathbb{N}$ be infinite sets such that $A+B$ contains all large integers. Let $A(x)=\lvert A\cap [1,x]\rvert$ and similarly for...
Let $C>0$. Does there exist a $c>0$ (depending on $C$) such that, for all sufficiently large $x$, if $A\subseteq [1,x]$ has $\sum_{n\in...
Fix some constant $C>0$ and let $n$ be large. Let $A\subseteq \{2,\ldots,n\}$ be such that $(a,b)=1$ for all $a\neq b\in A$ and $\sum_{n\in...
Do the squares contain arbitrarily long quasi-progressions? That is, does there exist some constant $C>0$ such that, for any $k$, the squares contain...
Let $f(k)$ be the minimal $n$ such that any $2$-colouring of $\{1,\ldots,n\}$ contains a monochromatic $k$-term descending wave: a sequence...
Suppose $n\geq kr+(t-1)(k-1)$ and the edges of the complete $r$-uniform hypergraph on $n$ vertices are $t$-coloured. Prove that some colour class...
Let $n> 1$ and $p_1<\cdots
Alice and Bob play a game on the edges of $K_n$, alternating colouring edges by red (Alice) and blue (Bob). Alice goes first, and wins if at the end...
If $\mathcal{F}$ is a family of subsets of $\{1,\ldots,n\}$ then we write $G_{\mathcal{F}}$ for the graph on $\mathcal{F}$ where $A\sim B$ if $A$ and...
Let $r\geq 2$ and $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ be such that $A_i\not\subseteq A_j$ for all $i\neq j$ and for any $t$ if there exists some...
Is there a $3$-uniform hypergraph on $n$ vertices which contains at least $n-O(1)$ different sizes of cliques (maximal complete subgraphs)
What is the size of the largest Sidon subset $A\subseteq\{1,2^2,\ldots,N^2\}$? Is it $N^{1-o(1)}$?
Let $k\geq 1$ and $H_k(n)$ be the maximal $r$ such that if $A\subset\mathbb{N}$ has $\lvert A\rvert=n$ and $\| 1_A\ast 1_A\|_\infty \leq k$ then $A$...
Let $f(n)$ be maximal such that, for every $m\geq 1$, there exists some $S\subseteq \{1,\ldots,n\}$ with $\lvert S\rvert=f(n)$ such that $m\neq...
Let $h(n)$ be minimal such that $2^n-1,3^n-1,\ldots,h(n)^n-1$ are mutually coprime.Does, for every prime $p$, the density $\delta_p$ of integers with...
Let $A\subset\mathbb{N}$ be the set of $n$ such that for every prime $p\mid n$ there exists some $d\mid n$ with $d>1$ such that $d\equiv 1\pmod{p}$....
Let $g_k(n)$ be the maximal number of edges possible on a graph with $n$ vertices which does not contain a cycle with $k$ chords incident to a vertex...
Let $f(n;k,l)=\min \mathrm{ex}(n;G)$, where $G$ ranges over all graphs with $k$ vertices and $l$ edges.Give good estimates for $f(n;k,l)$ in the...
Give an asymptotic formula for $\mathrm{ex}(n;C_4)$.
Let $A\subseteq \mathbb{N}$. Can there exist some constant $c>0$ such that\[\sum_{n\leq N} 1_A\ast 1_A\ast 1_A(n) = cN+O(1)?\]
Let $A\subseteq \mathbb{N}$. Can there exist some constant $c>0$ such that\[\sum_{n\leq N} 1_A\ast 1_A(n) = cN+O(1)?\]
The cochromatic number of $G$, denoted by $\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour...
The cochromatic number of $G$, denoted by $\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour...
The cochromatic number of $G$, denoted by $\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour...
Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\mathbb{R}^4$ in which every $x\in A$ has at least $f(n)$ points in $A$...
Let $G$ be a graph with minimum degree $k$ and girth $>2s$ (i.e. $G$ contains no cycles of length $\leq 2s$). Must there be $\gg k^s$ many distinct...
Let $G$ be a graph with chromatic number $\chi(G)=4$. If $m_1
Let $\epsilon>0$. Does there exist $A\subseteq \mathbb{N}$ such that the lower density of $A+A$ is at least $1-\epsilon$ and yet $1_A\ast 1_A(n)...
Let $f(n)$ count the number of sum-free $A\subseteq \{1,\ldots,n\}$, i.e. $A$ contains no solutions to $a=b+c$ with $a,b,c\in A$. Is it true...
How large should $\ell(n)$ be such that, almost surely, a random $3$-uniform hypergraph on $3n$ vertices with $\ell(n)$ edges must contain $n$...
Is it true that, almost surely, a random graph on $n$ vertices with $\geq (\tfrac{1}{2}+\epsilon)n\log n$ edges is Hamiltonian?
Describe the size of the second largest component of the random graph on $n$ vertices, where each edge is included independently with probability...
Let $k$ be a large fixed constant. Let $f_k(n)$ be the minimal $m$ such that there exists a graph $G$ on $n$ vertices with chromatic number $k$, such...
Let $T_2,\ldots,T_n$ be a collection of trees such that $T_k$ has $k$ vertices. Can we always write $K_n$ as the edge disjoint union of the $T_k$?
Let $G$ be a graph on $n$ vertices with diameter $2$, such that deleting any edge increases the diameter of $G$. Is it true that $G$ has at most...
Let $A\subseteq \mathbb{N}$ be such that $A+A$ has positive density. Can one always decompose $A=A_1\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$...
Let $\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\mathfrak{m}$. Let $r\geq 1$. Must $G$ contain a subgraph of...
Let $\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\mathfrak{m}$. Is it true that, for every infinite cardinal...
If $G$ has infinite chromatic number and is triangle-free (contains no $K_3$) then must $G$ contain every tree as an induced subgraph?
Let $G$ be a graph with chromatic number $\aleph_1$. Must there exist an edge $e$ such that, for all large $n$, $G$ contains a cycle of length $n$...
Let $G$ be a graph with chromatic number $\aleph_1$. Is there, for every cardinal number $m$, some graph $G_m$ of chromatic number $m$ such that...
Given any $n$ points in $\mathbb{R}^2$ when can one give positive weights to the points such that the sum of the weights of the points along every...
Find, for all large $n$, a non-trivial pairwise balanced block design $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ such that, for all $t$, there are...
Call a sequence $1
Call a sequence $1< X_1\leq \cdots \leq X_m\leq n$ block-compatible if there is a pairwise balanced block design $A_1,\ldots,A_m\subseteq...
Find some reasonable function $f(n)$ such that, for almost all integers $n$, the least integer $m$ such that $m\nmid \binom{2n}{n}$ satisfies\[m\sim...
Are there infinitely many pairs of integers $n\neq m$ such that $\binom{2n}{n}$ and $\binom{2m}{m}$ have the same set of prime divisors?
Let $C>0$ be a constant. Are there infinitely many integers $a,b,n$ with $a+b> n+C\log n$ such that the denominator of\[\frac{n!}{a!b!}\]contains...
Let $C>0$ and $\epsilon>0$ be sufficiently small. Are there infinitely many integers $a,b,n$ with $a\geq \epsilon n$ and $b\geq \epsilon n$ such...
Let $k\geq 2$. Does\[(n+k)!^2 \mid (2n)!\]for infinitely many $n$?
As $n\to \infty$ ranges over integers\[\sum_{p\leq n}1_{n\in (p/2,p)\pmod{p}}\frac{1}{p}\sim \frac{\log\log n}{2}.\]
Give an asymptotic formula for the number of $k\times n$Latin rectangles.
Let $f(n)$ be the maximum number ofmutually orthogonal Latin squaresof order $n$. Is it true that\[f(n) \gg n^{1/2}?\]
If there is afinite projective planeof order $n$ then must $n$ be a prime power?A finite projective plane of order $n$ is a collection of subsets of...
Let $k>r$ and $n$ be sufficiently large in terms of $k$ and $r$. Does there always exist a block $r-(n,k,1)$ design (orSteiner systemwith parameters...
Let $W(3,k)$ be the van der Waerden number defined as the minimum $n$ such that in any red/blue colouring of $\{1,\ldots,n\}$ there exists either a...
Let $\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any...
Let $\mathrm{ex}_r(n;K_{r+1}^r)$ be the maximum number of $r$-edges that can be placed on $n$ vertices without forming a $K_{r+1}^r$ (the $r$-uniform...
Is there some constant $C>0$ such that any graph on $n$ vertices with $\geq Cr^2n$ edges contains a subdivision of $K_r$?
Let $G$ be a graph on $n$ vertices with chromatic number $\chi(G)$ and let $\sigma(G)$ be the maximal $k$ such that $G$ contains a subdivision of...
Let $\mathcal{F}$ be the family of all $3$-uniform hypergraphs with $6$ vertices and $3$ $3$-edges. Is it true...
Does every regular graph of degree $4$ contain a regular subgraph of degree $3$? Is there any $r$ such that every regular graph of degree $r$ must...
Is it true that\[\mathrm{ex}(n; K_{r,r}) \gg n^{2-1/r}?\]
Is it true that, for every bipartite graph $G$, there exists some $\alpha\in [1,2)$ and $c>0$ such that\[\mathrm{ex}(n;G)\sim cn^\alpha?\]Must...
Determine, for any $k>r>2$, the value of\[\frac{\mathrm{ex}_r(n,K_k^r)}{\binom{n}{r}},\]where $\mathrm{ex}_r(n,K_k^r)$ is the largest number of...
Let $f(n,m)$ be minimal such that in $(m,m+f(n,m))$ there exist distinct integers $a_1,\ldots,a_n$ such that $k\mid a_k$ for all $1\leq k\leq n$....
Let $f(n)$ be minimal such that in $(n,n+f(n))$ there exist distinct integers $a_1,\ldots,a_n$ such that $k\mid a_k$ for all $1\leq k\leq n$. Obtain...
Let $f(n)$ be minimal such that, for any $A=\{a_1,\ldots,a_n\}\subseteq [2,\infty)\cap\mathbb{N}$ of size $n$, in any interval $I$ of $f(n)\max(A)$...
Let $g(n)$ be minimal such that for any $A\subseteq [2,\infty)\cap \mathbb{N}$ with $\lvert A\rvert =n$ and any set $I$ of $\max(A)$ consecutive...
Let $A\subset \mathbb{N}$ be a finite Sidon set. Is there some set $B$ with $A\subseteq B$ which isperfect difference setmodulo $p^2+p+1$ for some...
Let $L(r)$ be such that if $G$ is a graph formed by taking a finite set of points $P$ in $\mathbb{R}^2$ and some set $A\subset (0,\infty)$ of size...
Let $G$ be a finite unit distance graph in $\mathbb{R}^2$ (i.e. the vertices are a finite collection of points in $\mathbb{R}^2$ and there is an edge...
Let $G_n$ be the unit distance graph in $\mathbb{R}^n$, with two vertices joined by an edge if and only if the distance between them is $1$.Estimate...
Let $r\geq 1$ and define $T(n,r)$ to be maximal such that there exists a family $\mathcal{F}$ of subsets of $\{1,\ldots,n\}$ of size $T(n,r)$ such...
Let $k\geq 4$. If $\mathcal{F}$ is a family of subsets of $\{1,\ldots,n\}$ with $\lvert A\rvert=k$ for all $A\in \mathcal{F}$ and $\lvert...
Let $\mathcal{F}$ be a family of sets closed under taking subsets (i.e. if $B\subseteq A\in\mathcal{F}$ then $B\in \mathcal{F}$). There exists some...
Let\[f(n)=\min_{1
Is it true that for every $1\leq i
Is there some $h(n)\to \infty$ such that for all $2\leq i
Let $\delta(m,\alpha)$ denote the density of the set of integers which are divisible by some $d\equiv 1\pmod{m}$ with $1
Let $h(n)$ be the largest $\ell$ such that there is a sequence of primes $p_1<\cdots < p_\ell$ all dividing $n$ with $p_{i+1}\equiv 1\pmod{p_i}$. Let...
Let $p_1
Let $f_{\max}(n)$ be the largest $m$ such that $\phi(m)=n$, and $f_{\min}(n)$ be the smallest such $m$, where $\phi$ is Euler's totient function....
Let $k\geq 2$ and $n$ be sufficiently large depending on $k$. Let $A=\{a_1
Let $\delta_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$. Is $\delta_1(n,m)$ unimodular for $m>n+1$ (i.e....
Given $A\subseteq \mathbb{N}$ let $M_A=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}$ be the set of multiples of $A$. Find a necessary and...
Let $d_k(p)$ be the density of those integers whose $k$th smallest prime factor is $p$ (i.e. if $p_1
Let $n$ be sufficiently large. Is there some choice of congruence class $a_p$ for all primes $2\leq p\leq n$ such that every integer in $[1,n]$...
Define $\epsilon_n$ to be maximal such that there exists some choice of congruence class $a_p$ for all primes $n^{\epsilon_n}
Let $Y(x)$ be the maximal $y$ such that there exists a choice of congruence classes $a_p$ for all primes $p\leq x$ such that every integer in $[1,y]$...
Can every integer $N\geq 2$ be written as\[N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}\]for some $k\geq 2$ and $m\geq n+k$?
Let $\epsilon>0$ and $n$ be large depending on $\epsilon$. Is it true that for all $n^\epsilon
For $0\leq k\leq n$ write\[\binom{n}{k} = uv\]where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $(k,n]$.Let...
Is it true that for every $1\leq k\leq n$ the largest prime divisor of $\binom{n}{k}$, say $P(\binom{n}{k})$,...
Is it true that for almost all $n$ there exists some $m\in (p_n,p_{n+1})$ such that\[p(m) \geq p_{n+1}-p_n,\]where $p(m)$ denotes the least prime...
Is it true that for all large $n$ there exists $k$ such that $n+k$ is composite and\[p(n+k)>k^2,\]where $p(m)$ is the least prime factor of $m$?
Is it true that, for all sufficiently large $n$, there exists some $k$ such that\[p(n+k)>k^2+1,\]where $p(m)$ denotes the least prime factor of...
Let $\epsilon>0$ and $\omega(n)$ count the number of distinct prime factors of $n$. Are there infinitely many values of $n$ such that\[\omega(n-k) <...
Let $M(n,k)=[n+1,\ldots,n+k]$ be the least common multiple of $\{n+1,\ldots,n+k\}$.Are there infinitely many $m,n$ and $k\geq 3$ with $m\geq n+k$...
Let $M(n,k)=[n+1,\ldots,n+k]$ be the least common multiple of $\{n+1,\ldots,n+k\}$.Is it true that for all $m\geq n+k$\[M(n,k) \neq M(m,k)?\]
Is every sufficiently large integer of the form\[ap^2+b\]for some prime $p$ and integer $a\geq 1$ and $0\leq b
We say that $A\subset \mathbb{N}$ has the translation property if, for every $n$, there exists some integer $t_n\geq 1$ such that, for all $1\leq...
Are there any integer solutions to $x^xy^y=z^z$ with $x,y,z>1$?
Let $1=d_1<\cdots
Can the product of an arithmetic progression of positive integers $n,n+d,\ldots,n+(k-1)d$ of length $k\geq 4$ (with $(n,d)=1$) be a perfect power?
Given $a_{i}^n\in [-1,1]$ for all $1\leq i\leq n<\infty$ we define $p_{i}^n$ as the unique polynomial of degree $n-1$ such that $p_{i}^n(a_{i}^n)=1$...
Let $A\subseteq \mathbb{R}^d$ be a set of $n$ points such that all pairwise distances differ by at least $1$. Is the diameter of $A$ at least...
Let $F_k(n)$ be minimal such that for any $n$ points in $\mathbb{R}^2$ there exist at most $F_k(n)$ many distinct lines passing through at least $k$...
Is it true that the number of incongruent sets of $n$ points in $\mathbb{R}^2$ which maximise the number of unit distances tends to infinity as...
Let $p,q\geq 1$ be fixed integers. We define $H(n)=H(N;p,q)$ to be the largest $m$ such that any graph on $n$ vertices where every set of $p$...
Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that, for every $\epsilon>0$, if...
Is there some constant $c$ such that for every $n$ there are $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ such that $\lvert A_i\rvert >n^{1/2}-c$ for all...
Let $c<1$ be some constant and $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ be such that $\lvert A_i\rvert >c\sqrt{n}$ for all $i$ and $\lvert A_i\cap...
Let $k\geq 2$ and $q(n,k)$ denote the least prime which does not divide $\prod_{1\leq i\leq k}(n+i)$. Is it true that, if $k$ is fixed and $n$ is...
Consider the triangular lattice with minimal distance between two points $1$. Denote by $f(t)$ the number of distances from any points $\leq t$. For...
Are there, for all large $n$, some points $x_1,\ldots,x_n,y_1,\ldots,y_n\in \mathbb{R}^2$ such that the number of distinct distances $d(x_i,y_j)$...
Let $x_1,\ldots,x_n\in \mathbb{R}^3$ be the vertices of a convex polyhedron. Are there at least\[(1-o(1))\frac{n}{2}\]many distinct distances between...
Is there a set of $n$ points in $\mathbb{R}^2$ such that every subset of $4$ points determines at least $3$ distances, yet the total number of...
Let $\delta>0$ and $N$ be sufficiently large depending on $\delta$. Is it true that if $A\subseteq \{1,\ldots,N\}^2$ has $\lvert A\rvert \geq \delta...
Is it true that if $A\subset \mathbb{R}^2$ is a set of $n$ points such that every subset of $3$ points determines $3$ distinct distances (i.e. $A$...
Let $A\subset \mathbb{N}$ be a set with positive upper density. Must there exist an infinite set $B$ and integer $t$ such that\[B+B+t\subseteq A?\]
Let $x_1,\ldots,x_n\in \mathbb{R}^2$ be such that no circle whose centre is one of the $x_i$ contains three other points. Are there at...
Let $x_1,\ldots,x_n\in \mathbb{R}^2$ with no four points on a circle. Must there exist some $x_i$ with at least $(1-o(1))n$ distinct distances to...
Let $x_1,\ldots,x_n\in \mathbb{R}^2$ and let $R(x_i)=\#\{ \lvert x_j-x_i\rvert : j\neq i\}$, where the points are ordered such that\[R(x_1)\leq...
Let $x_1,\ldots,x_n\in \mathbb{R}^2$ and let $R(x_i)=\#\{ \lvert x_j-x_i\rvert : j\neq i\}$, where the points are ordered such that\[R(x_1)\leq...
Let $f_k(n)$ denote the smallest integer such that any $f_k(n)$ points in general position in $\mathbb{R}^k$ contain $n$ which determine a convex...
Let $f(m)$ be such that if $A\subseteq \{1,\ldots,N\}$ has $\lvert A\rvert=m$ then every interval in $[1,\infty)$ of length $2N$ contains $\geq f(m)$...
Let $P(m)$ denote the greatest prime factor of $m$. Is it true that, for any two primes $p,q$, there exists some integer $n$ such that $P(n)=p$ and...
Let $g(n)$ denote the largest $t$ such that there exist integers $2\leq a_1
Let $\tau(n)$ count the number of divisors of $n$. Is there some $n>24$ such that\[\max_{m
Let $p_1,\ldots,p_k$ be distinct primes. Are there infinitely many $n$ such that $n!$ is divisible by an even power of each of the $p_i$?
If $\mathbb{N}$ is 2-coloured then must there exist a monochromatic three-term arithmetic progression $x,x+d,x+2d$ such that $d>x$?
Let $f(k,r)$ be minimal such that if $A_1,A_2,\ldots$ is a family of sets, all of size $k$, such that for every collection of $r$ of the $A_is$ there...
Let $f(n;t)$ be minimal such that if a $t$-uniform hypergraph on $n$ vertices contains at least $f(n;t)$ edges then there must be four edges...
Let $f(n)$ be the maximal number of edges in a graph on $n$ vertices such that all cycles have more vertices than diagonals. Is it true that $f(n)\ll...
Is there some function $f$ such that for all $k\geq 1$ if a finite graph $G$ has chromatic number $\geq f(k)$ then $G$ has $k$ edge disjoint cycles...
Is there some function $f$ such that for all $k\geq 3$ if a finite graph $G$ has chromatic number $\geq f(k)$ then $G$ must contain some odd cycle...
Is it true that if the edges of $K_n$ are 2-coloured then there are at most $n^2/4$ many edges which do not occur in a monochromatic triangle?
Let $S$ be a family of finite graphs such that for every $n$ there is some $G_n\in S$ such that if the edges of $G_n$ are coloured with $n$ colours...
If $G$ is a graph on $n$ vertices which contains no complete graph or independent set on $\gg \log n$ vertices then $G$ contains an induced subgraph...
Suppose $G$ is a graph on $n$ vertices which contains no complete graph or independent set on $\gg \log n$ many vertices. Must $G$ contain $\gg...
Let $t\geq 1$ and $A\subseteq \{1,\ldots,N\}$ be such that whenever $a,b\in A$ with $b-a\geq t$ we have $b-a\nmid b$. How large can $\lvert A\rvert$...
Find all $n$ such that there is at least one triangle which can be cut into $n$ congruent triangles.
Classify those triangles which can only be cut into a square number of congruent triangles.
A graph is $(a,b)$-choosable if for any assignment of a list of $a$ colours to each of its vertices there is a subset of $b$ colours from each list...
The list chromatic number $\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$...
The list chromatic number $\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$...
The list chromatic number $\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$...
Let $G$ be a graph with chromatic number $k$ containing no $K_k$. If $a,b\geq 2$ and $a+b=k+1$ then must there exist two disjoint subgraphs of $G$...
Let $\omega(G)$ denote the clique number of $G$ and $\chi(G)$ the chromatic number. If $f(n)$ is the maximum value of $\chi(G)/\omega(G)$, as $G$...
Let $k\geq 4$ and $g_k(n)$ denote the largest $m$ such that there is a graph on $n$ vertices with chromatic number $k$ and girth $>m$ (i.e. contains...
The cochromatic number of $G$, denoted by $\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour...
Let $X$ be a finite set of size $n$ and $H(n)$ be such that there is a function $f:\{A : A\subseteq X\}\to X$ so that for every $Y\subseteq X$ with...
Let $X$ be a set of cardinality $\aleph_\omega$ and $f$ be a function from the finite subsets of $X$ to $X$ such that $f(A)\not\in A$ for all $A$....
Let $G$ be a regular graph with $2n$ vertices and degree $n+1$. Must $G$ have $\gg 2^{2n}$ subsets that are spanned by a cycle?
Let $G$ be a graph on $n$ vertices, $\alpha_1(G)$ be the maximum number of edges that contain at most one edge from every triangle, and $\tau_1(G)$...
If $G$ is a graph on $n$ vertices without a $K_4$ then how large a triangle-free induced subgraph must $G$ contain?
For a triangle-free graph $G$ let $h_r(G)$ be the smallest number of edges that need to be added to $G$ so that it has diameter $r$ (while preserving...
For a triangle-free graph $G$ let $h_2(G)$ be the smallest number of edges that need to be added to $G$ so that it has diameter $2$ and is still...
Let $r\geq 3$. If the edges of $K_{r^2+1}$ are $r$-coloured then there exist $r+1$ vertices with at least one colour missing on the edges of the...
Let $r\geq 3$. For an $r$-uniform hypergraph $G$ let $\tau(G)$ denote the covering number (or transversal number), the minimum size of a set of...
Does there exist some constant $c>0$ such that if $G$ is a graph with $n$ vertices and $\geq (1/8-c)n^2$ edges then $G$ must contain either a $K_4$...
Let $f(n,k)$ be minimal such that there is a graph with $n$ vertices and $f(n,k)$ edges where every set of $k+2$ vertices induces a subgraph with...
Let $n\geq 3$ and $G$ be a graph with $\binom{2n+1}{2}-\binom{n}{2}-1$ edges. Must $G$ be the union of a bipartite graph and a graph with maximum...
Let $G$ be a connected graph with $n$ vertices, minimum degree $d$, and diameter $D$. Show if that $G$ contains no $K_{2r}$ and $(r-1)(3r+2)\mid d$...
For a graph $G$ let $\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes called the...
For a graph $G$ let $\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes called the...
Let $f(n)$ be the minimal $m$ such that if the edges of $K_{2^n+1}$ are coloured with $n$ colours then there must be a monochromatic odd cycle of...
Let $G$ be a graph with $n$ vertices and $>n^2/4$ many edges. Are there at least $\frac{2}{9}n^2$ edges of $G$ which are contained in a $C_5$?
For a set of $n$ points $P\subset \mathbb{R}^2$ let $\ell_1,\ldots,\ell_m$ be the lines determined by $P$, and let $A=\{\lvert \ell_1\cap...
Given any $n$ distinct points in $\mathbb{R}^2$ let $f(n)$ count the number of distinct lines determined by these points. What are the possible...
Is there some function $f(n)\to \infty$ as $n\to\infty$ such that there exist $n$ distinct points on the surface of a two-dimensional sphere with at...
Given $n$ distinct points $A\subset\mathbb{R}^2$ must there be a point $x\in A$ such that\[\#\{ d(x,y) : y \in A\} \gg n^{1-o(1)}?\]Or even $\gg...
Let $(A_i)$ be a family of countably infinite sets such that $\lvert A_i\cap A_j\rvert \neq 2$ for all $i\neq j$. Find the smallest cardinal $C$ such...
Let $(A_i)$ be a family of sets with $\lvert A_i\rvert=\aleph_0$ for all $i$, such that for any $i\neq j$ we have $\lvert A_i\cap A_j\rvert$ finite...
For which limit ordinals $\alpha$ is it true that if $G$ is a graph with vertex set $\alpha$ then $G$ must have either an infinite path or...
Let $e(n,r)$ be minimal such that every graph on $n$ vertices with at least $e(n,r)$ edges, each edge contained in at least one triangle, must have...
Let $G$ be a (possibly infinite) graph and $A,B$ be disjoint independent sets of vertices. Must there exist a family $P$ of disjoint paths between...
Let $m$ be an infinite cardinal and $\kappa$ be the successor cardinal of $2^{\aleph_0}$. Can one colour the countable subsets of $m$ using $\kappa$...
Let $G$ be a graph on at most $\aleph_1$ vertices which contains no $K_4$ and no $K_{\aleph_0,\aleph_0}$ (the complete bipartite graph with...
For which graphs $G_1,G_2$ is it true thatfor every $n\geq 1$ there is a graph $H$ without a $G_1$ but if the edges of $H$ are $n$-coloured then...
Is there an infinite graph $G$ which contains no $K_4$ and is not the union of countably many triangle-free graphs?
Does every graph $G$ with chromatic number $\geq \aleph_1$ contain all sufficiently large odd cycles?
Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number $>\aleph_0$.
Determine which countable ordinals $\beta$ have the property that, if $\alpha=\omega^{^\beta}$, then in any red/blue colouring of the edges of...
Let $\alpha$ be the infinite ordinal $\omega^{\omega^2}$. Is it true that in any red/blue colouring of the edges of $K_\alpha$ there is either a red...
Let $\alpha$ be the infinite ordinal $\omega^\omega$. Is it true that in any red/blue colouring of the edges of $K_\alpha$ there is either a red...
Let $g(n)$ be maximal such that in any set of $n$ points in $\mathbb{R}^2$ with no four points on a line there exists a subset on $g(n)$ points with...
Let $f_k(n)$ be minimal such that if $n$ points in $\mathbb{R}^2$ have no $k+1$ points on a line then there must be at most $f_k(n)$ many lines...
What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that, for all $\emptyset\neq S\subseteq A$, $\sum_{n\in S}n$ is not a square?
Is there a covering system such that no two of the moduli divide each other?
What is the maximum number of edges that a graph on $n$ vertices can have if it does not contain two edge-disjoint cycles with the same vertex set?
Let $G$ be a graph with $n$ vertices and $\delta n^{2}$ edges. Are there subgraphs $H_1,H_2\subseteq G$ such that$H_1$ has $\gg \delta^3n^2$ edges...
Every connected graph on $n$ vertices can be partitioned into at most $\lceil n/2\rceil$ edge-disjoint paths.
Does there exist a graph $G$ which contains no $K_4$, and yet any $2$-colouring of the edges produces a monochromatic $K_3$?
Let $f(m)$ be the maximal $k$ such that a triangle-free graph on $m$ edges must contain a bipartite graph with $k$ edges. Determine $f(m)$.
Let $G$ be a graph on $n$ vertices such that at least $n/2$ vertices have degree at least $n/2$. Must $G$ contain every tree on at most $n/2$...
Let $\delta>0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_{2,2,2}$ and at least $\delta n^2$ edges then $G$ contains...
If $G$ is a random graph on $2^d$ vertices, including each edge with probability $1/2$, then $G$ almost surely contains a copy of $Q_d$ (the...
If $G$ is a graph with $4k$ vertices and minimum degree at least $2k$ then $G$ contains $k$ vertex-disjoint $4$-cycles.
Let $Q_k$ be the $k$-dimensional hypercube graph (so that $Q_k$ has $2^k$ vertices and $k2^{k-1}$ edges). Determine the behaviour...
If $\mathcal{F}$ is a finite set of finite graphs then $\mathrm{ex}(n;\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have...
Is it true that, for $k\geq 2$,\[\mathrm{ex}(n;\{C_{2k-1},C_{2k}\})=(1+o(1))(n/2)^{1+\frac{1}{k}}.\]
Is it true that\[\mathrm{ex}(n;\{C_3,C_4\})\sim (n/2)^{3/2}?\]
Show that for $k\geq 3$\[\mathrm{ex}(n;C_{2k})\gg n^{1+\frac{1}{k}}.\]
Show that for any rational $\alpha \in [1,2)$ there exists a bipartite graph $G$ such that\[\mathrm{ex}(n;G)\asymp n^{\alpha}.\]
Let $k\geq 3$. Is it true that, for any graph $H$ on $m$ edges without isolated vertices,\[R(C_k,H) \leq 2m+\left\lceil\frac{k-1}{2}\right\rceil?\]
Let $k\geq 1$. What is the best possible $c_k$ such that\[R(C_{2k+1},H)\leq c_k m\]for any graph $H$ on $m$ edges without isolated vertices?
Let $G$ be a graph such that $R(G,T_n)\ll n$ for any tree $T_n$ on $n$ vertices and $R(G,K_n)\ll n^2$. Is it true that, for any $H$ with $m$ edges...
Let $G$ be either $Q_3$ or $K_{3,3}$ or $H_5$ (the last formed by adding two vertex-disjoint chords to $C_5$). Is it true that, if $H$ has $m$ edges...
Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges. Is it true that, if $H$ has $m$ edges and no isolated vertices,...
Let $R^*(G)$ be the induced Ramsey number: the minimal $m$ such that there is a graph $H$ on $m$ vertices such that any $2$-colouring of the edges of...
Let $R_3(n)$ be the minimal $m$ such that if the edges of the $3$-uniform hypergraph on $m$ vertices are $2$-coloured then there is a monochromatic...
Let $F(n,\alpha)$ denote the largest $m$ such that there exists a $2$-colouring of the edges of $K_n$ so that every $X\subseteq [n]$ with $\lvert...
Let $R_r(n)$ denote the $r$-uniform hypergraph Ramsey number: the minimal $m$ such that if we $2$-colour all edges of the complete $r$-uniform...
Let $\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any...
Let $\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any...
Let $\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges that is Ramsey for...
Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$....
Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Is it true...
Let $R(G;3)$ denote the minimal $m$ such that if the edges of $K_m$ are $3$-coloured then there must be a monochromatic copy of $G$. Show...
Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine the value...
Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Show that\[\lim_{k\to...
Let $R(3,3,n)$ denote the smallest integer $m$ such that if we $3$-colour the edges of $K_m$ then there is either a monochromatic triangle in one of...
Determine the Ramsey number\[R(C_4,S_n),\]where $S_n=K_{1,n}$ is the star on $n+1$ vertices.In particular, is it true that, for any $c>0$, there are...
Prove that\[R(C_k,K_n)=(k-1)(n-1)+1\]for $k\geq n\geq 3$ (except when $n=k=3$).
Let $m_1\leq\cdots\leq m_k$ and $n$ be sufficiently large. If $T$ is a tree on $n$ vertices and $G$ is the complete multipartite graph with vertex...
If $T$ is a tree which is a bipartite graph with $k$ vertices and $2k$ vertices in the other class then\[R(T)=4k-1.\]
Let $n\geq k+1$. Every graph on $n$ vertices with at least $\frac{k-1}{2}n+1$ edges contains every tree on $k+1$ vertices.
If $T$ is a tree on $n$ vertices then\[R(T) \leq 2n-2.\]
Let $G$ be a graph with no isolated vertices and $m$ edges. Is it true that\[R(G) \leq 2^{O(m^{1/2})}?\]
Let $G$ be a graph with $m$ edges and no isolated vertices. Is the Ramsey number $R(G)$ maximised when $G$ is 'as complete as possible'? That is, if...
Show that\[R(3,k+1)-R(3,k)\to\infty\]as $k\to \infty$. Similarly, prove or disprove that\[R(3,k+1)-R(3,k)=o(k).\]
Define $f(N)$ be the minimal $k$ such that the following holds: if $G$ is an abelian group of size $N$ and $A\subseteq G$ is a random set of size $k$...
Is it true that if $A\subseteq\{1,\ldots,n\}$ is a set such that $[a,b]>n$ for all $a\neq b$, where $[a,b]$ is the least common multiple,...
Let $a_1,\ldots,a_p$ be (not necessarily distinct) residues modulo $p$, such that there exists some $r$ so that if $S\subseteq [p]$ is non-empty...
Is it true that if $A\subseteq \mathbb{Z}/N\mathbb{Z}$ has size $\gg N^{1/2}$ then there exists some non-empty $S\subseteq A$ such that $\sum_{n\in...
Let $h(n)$ be such that, for any set $A\subseteq \mathbb{N}$ of size $n$, the set\[\left\{ \frac{a}{(a,b)}: a,b\in A\right\}\]has size at least...
Let $r\geq 2$ and suppose that $A\subseteq\{1,\ldots,N\}$ is such that, for any $m$, there are at most $r$ solutions to $m=pa$ where $p$ is prime and...
Let $\epsilon>0$ and $N$ be sufficiently large. If $A\subseteq \{1,\ldots,N\}$ has $\lvert A\rvert \geq \epsilon N$ then must there exist...
Let $\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\subseteq \{1,\ldots,N\}$ has size at least $\epsilon N$ then there must be...
Let $r\geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\{1,\ldots,N\}$ such that no subset of size $r$ has the same pairwise...
What is the largest possible subset $A\subseteq\{1,\ldots,N\}$ which contains $N$ such that $\mathrm{gcd}(a,b)>1$ for all $a\neq b\in A$?
Let $\delta>0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_5$ and at least $\delta n^2$ edges then $G$ contains a set...
If $\mathbb{N}$ is 2-coloured then is there some infinite set $A\subseteq \mathbb{N}$ such that all finite subset sums\[ \sum_{n\in S}n\](as $S$...
Let $F(k)$ be the minimal $N$ such that if we two-colour $\{1,\ldots,N\}$ there is a set $A$ of size $k$ such that all subset sums $\sum_{a\in S}a$...
Let $\ell(N)$ be maximal such that in any finite set $A\subset \mathbb{R}$ of size $N$ there exists a Sidon subset $S$ of size $\ell(N)$ (i.e. the...
Let $d_k(n)$ be the expected distance from the origin after taking $n$ random steps from the origin in $\mathbb{Z}^k$ (conditional on no self...
Let $f(n,k)$ count the number ofself-avoiding walksof $n$ steps (beginning at the origin) in $\mathbb{Z}^k$ (i.e. those walks which do not intersect...
Let $a_n\in \mathbb{R}$ be such that $\sum_n \lvert a_n\rvert^2=\infty$ and $\lvert a_n\rvert=o(1/\sqrt{n})$. Is it true that, for almost all...
Let $a_n\geq 0$ with $a_n\to 0$ and $\sum a_n=\infty$. Find a necessary and sufficient condition on the $a_n$ such that, if we choose (independently...
Is it true that all except at most $o(2^n)$ many degree $n$ polynomials with $\pm 1$-valued coefficients $f(z)$ have $\lvert f(z)\rvert <1$ for some...
For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). What is the correct order of magnitude (for...
Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{-1,1\}$ independently uniformly at random for $0\leq...
Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{-1,1\}$ independently uniformly at random for $0\leq...
Let $(\epsilon_k)_{k\geq 0}$ be independently uniformly chosen at random from $\{-1,1\}$. If $R_n$ counts the number of real roots of...
Let $f$ be a Rademacher multiplicative function: a random $\{-1,0,1\}$-valued multiplicative function, where for each prime $p$ we independently...
Let $z_1,\ldots,z_n\in \mathbb{C}$ with $z_1=1$. Must there exist an absolute constant $c>0$ such that\[\max_{1\leq k\leq n}\left\lvert...
Is it true that, in any two-colouring of the edges of $K_n$, there exist $\sqrt{n}$ monochromatic paths, all of the same colour, which cover all...
Let $f(z)=\sum_{k=1}^\infty a_kz^{n_k}$ be an entire function (with $a_k\neq 0$ for all $k\geq 1$). Is it true that if $n_k/k\to \infty$ then $f(z)$...
Let $f(z)=\sum_{k\geq 1}a_k z^{n_k}$ be an entire function of finite order such that $\lim n_k/k=\infty$. Let $M(r)=\max_{\lvert z\rvert=r}\lvert...
Let $f(z)$ be an entire function, not a polynomial. Does there exist a locally rectifiable path $C$ tending to infinity such that, for every...
Let $f(z)$ be an entire function. Does there exist a path $L$ so that, for every $n$,\[\lvert f(z)/z^n\rvert \to \infty\]as $z\to \infty$ along...
Let $f=\sum_{n=0}^\infty a_nz^n$ be a transcendental entire function. What is the greatest possible value of\[\liminf_{r\to \infty}...
Is it true that, if $A\subset \mathbb{Z}$ is a finite set of size $N$, then\[\int_0^1 \left\lvert \sum_{n\in A}e(n\theta)\right\rvert...
Let $f(z)\in \mathbb{C}[z]$ be a monic polynomial of degree $n$. Is it true that, for every $c>1$, the set\[\{ z\in \mathbb{C} : \lvert f(z)\rvert<...
If $A\subset \mathbb{Z}$ is a finite set of size $N$ then is there some absolute constant $c>0$ and $\theta$ such that\[\sum_{n\in A}\cos(n\theta) <...
Let $f(z)\in\mathbb{C}[z]$ be a monic non-constant polynomial. Can the set\[\{ z\in \mathbb{C} : \lvert f(z)\rvert \leq 1\}\]be covered by a set of...
What is the chromatic number of the plane? That is, what is the smallest number of colours required to colour $\mathbb{R}^2$ such that no two points...
Let $\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which determine a triangle of area at most $\alpha(n)$....
What is the minimum number of circles determined by any $n$ points in $\mathbb{R}^2$, not all on a circle?
Is every set of diameter $1$ in $\mathbb{R}^n$ the union of at most $n+1$ sets of diameter $<1$?
Let $\alpha_n$ be the supremum of all $0\leq \alpha\leq \pi$ such that in every set $A\subset \mathbb{R}^2$ of size $n$ there exist three distinct...
What is the size of the largest $A\subseteq \mathbb{R}^d$ such that every three points from $A$ determine an isosceles triangle? That is, for any...
What is the size of the largest $A\subseteq \mathbb{R}^n$ such that there are only two distinct distances between elements of $A$? That is,\[\# \{...
For every $x\in\mathbb{R}$ let $A_x\subset \mathbb{R}$ be a bounded set with outer measure $<1$. Must there exist an infinite independent set, that...
What is $\mathrm{ex}_3(n,K_4^3)$? That is, the largest number of $3$-edges which can placed on $n$ vertices so that there exists no $K_4^3$, a set of...
Let $M=(a_{ij})$ be a real $n\times n$ doubly stochastic matrix (i.e. the entries are non-negative and each column and row sums to $1$). Does there...
Let $z_1,\ldots,z_n\in\mathbb{C}$ with $1\leq \lvert z_i\rvert$ for $1\leq i\leq n$. Let $D$ be an arbitrary disc of radius $1$. Is it true that the...
How many antichains in $[n]$ are there? That is, how many families of subsets of $[n]$ are there such that, if $\mathcal{F}$ is such a family and...
Let $\alpha \in \mathbb{R}$ be irrational and $\epsilon>0$. Are there positive integers $x,y,z$ such that\[\lvert x^2+y^2-z^2\alpha\rvert <\epsilon?\]
Let $\alpha,\beta \in \mathbb{R}$. Is it true that\[\liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0\]where $\|x\|$ is the distance from $x$ to...
If $A\subset \mathbb{C}$ is a finite set and $k\geq 1$ then let\[A_k = \{ z_1+\cdots+z_k : z_i\in A\textrm{ distinct}\}.\]For $k>2$ does the multiset...
Does there exist a $k$ such that every sufficiently large integer can be written in the form\[\prod_{i=1}^k a_i - \sum_{i=1}^k a_i\]for some integers...
Let $A=\{a_1
Let $f:\mathbb{N}\to \mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b)=1$). If there is a constant $c$ such that $\lvert...
Let $A,B\subseteq \{1,\ldots,N\}$ be such that all the products $ab$ with $a\in A$ and $b\in B$ are distinct. Is it true that\[\lvert A\rvert \lvert...
Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap [1,x]\rvert=o(x^{1/2})$. Let\[B=\{ n\geq 1 : a\nmid n\textrm{ for all }a\in A\}.\]If...
Let $A$ be a finite set and\[B=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}.\]Is it true that, for every $m>n\geq \max(A)$,\[\frac{\lvert B\cap...
Let $A\subseteq \mathbb{N}$ have positive density. Must there exist distinct $a,b,c\in A$ such that $[a,b]=c$ (where $[a,b]$ is the least common...
Let $A\subseteq \mathbb{N}$, and for each $n\in A$ choose some $X_n\subseteq \mathbb{Z}/n\mathbb{Z}$. Let\[B = \{ m\in \mathbb{N} : m\not\in...
Let $f(k)$ be the minimum number of terms in $P(x)^2$, where $P\in \mathbb{Q}[x]$ ranges over all polynomials with exactly $k$ non-zero terms. Is it...
Prove that there exists an absolute constant $c>0$ such that, whenever $\{1,\ldots,N\}$ is $k$-coloured (and $N$ is large enough depending on $k$)...
Let $f(k)$ be the minimal $N$ such that if $\{1,\ldots,N\}$ is $k$-coloured then there is a monochromatic solution to $a+b=c$. Estimate $f(k)$. In...
Define a sequence by $a_1=1$ and\[a_{n+1}=\lfloor\sqrt{2}(a_n+1/2)\rfloor\]for $n\geq 1$. The difference $a_{2n+1}-2a_{2n-1}$ is the $n$th digit in...
Let $a_1,\ldots,a_r,b_1,\ldots,b_r\in \mathbb{N}$ such that $\sum_{i}\frac{1}{a_i}>1$. For any finite sequence of $n$ (not necessarily distinct)...
Let $x_1,x_2,\ldots\in [0,1]$ be an infinite sequence. Is it true that\[\inf_n \liminf_{m\to \infty} n \lvert x_{m+n}-x_m\rvert\leq 5^{-1/2}\approx...
Is it true that, for all $k\neq 1$, there are infinitely many $n$ such that $2^n\equiv k\pmod{n}$?
Let $p$ be a prime and\[A_p = \{ k! \pmod{p} : 1\leq k
Is there a polynomial $f:\mathbb{Z}\to \mathbb{Z}$ of degree at least $2$ and a set $A\subset \mathbb{Z}$ such that for any $n\in \mathbb{Z}$ there...
Let $A\subseteq \mathbb{F}_p$. Let\[A\hat{+}A = \{ a+b : a\neq b \in A\}.\]Is it true that\[\lvert A\hat{+}A\rvert \geq \min(2\lvert A\rvert-3,p)?\]
Let $p$ be a prime. Given any finite set $A\subseteq \mathbb{F}_p\backslash \{0\}$, is there always a rearrangement $A=\{a_1,\ldots,a_t\}$ such that...
Under what set theoretic assumptions is it true that $\mathbb{R}^2$ can be $3$-coloured such that, for every uncountable $A\subseteq \mathbb{R}^2$,...
Is there a permutation $a_1,a_2,\ldots$ of the positive integers such that $a_k+a_{k+1}$ is always prime?
Given some initial finite sequence of primes $q_1<\cdots
Given a finite set of primes $Q=Q_0$, define a sequence of sets $Q_i$ by letting $Q_{i+1}$ be $Q_i$ together with all primes formed by adding three...
Call $n$weirdif $\sigma(n)\geq 2n$ and $n$ is not pseudoperfect, that is, it is not the sum of any set of its divisors.Are there any odd weird...
Let $A$ be the set of all $n$ such that $n=d_1+\cdots+d_k$ with $d_i$ distinct proper divisors of $n$, but this is not true for any $m\mid n$ with...
For any $n$ let $D_n$ be the set of sums of the shape $d_1,d_1+d_2,d_1+d_2+d_3,\ldots$ where $1
Prove the following for all large $x$: there is a choice of congruence classes $a_p$ for all primes $p\leq x$ and a decomposition $\{p\leq...
Let $N(X,\delta)$ denote the maximum number of points $P_1,\ldots,P_n$ which can be chosen in a circle of radius $X$ such that\[\| \lvert...
Let $N(X,\delta)$ denote the maximum number of points $P_1,\ldots,P_n$ which can be chosen in a circle of radius $X$ such that\[\| \lvert...
Let $A=\{n_1
Is there a function $f$ with $f(n)\to \infty$ as $n\to \infty$ such that, for all large $n$, there is a composite number $m$ such...
Let $p(n)$ denote the least prime factor of $n$. There is a constant $c>0$ such that\[\sum_{\substack{n
Let $s_t(n)$ be the $t$-smooth component of $n$ - that is, the product of all primes $p$ (with multiplicity) dividing $n$ such that $p
Let $a_0=0$ and $a_1=1$, and in general define $a_k$ to be the least integer $>a_{k-1}$ for which $(n-a_k,n-a_i)=1$ for all $0\leq i
Let $f(u)$ be the largest $v$ such that no $m\in (u,v)$ is composed entirely of primes dividing $uv$. Estimate $f(u)$.
Let $[1,\ldots,n]$ denote the least common multiple of $\{1,\ldots,n\}$. Is it true that, for all $k\geq 1$,\[[1,\ldots,p_{k+1}-1]<...
Is there some $\epsilon>0$ such that there are infinitely many $n$ where all primes $p\leq (2+\epsilon)\log n$ divide\[\prod_{1\leq i\leq \log...
Let $p_n$ be the smallest prime $\equiv 1\pmod{n}$ and let $m_n$ be the smallest integer such that $n\mid \phi(m_n)$.Is it true that $m_n
Let $q_1
Let\[f(n) = \min_{i
Is it true that, for all sufficiently large $n$, there exists some $i
Let $\omega(n)$ count the number of distinct prime factors of $n$. What is the size of the largest interval $I\subseteq [x,2x]$ such that...
Estimate $n_k$, the smallest integer $>2k$ such that $\prod_{1\leq i\leq k}(n_k-i)$ has no prime factor in $(k,2k)$.
How large must $y=y(\epsilon,n)$ be such that the number of integers in $(x,x+y)$ with a divisor in $(n,2n)$ is at most $\epsilon y$?
Let $r(n)$ count the number of $d_1,d_2$ such that $d_1\mid n$ and $d_2\mid n$ and $d_1
Let $\tau(n)$ count the divisors of $n$ and $\tau^+(n)$ count the number of $k$ such that $n$ has a divisor in $[2^k,2^{k+1})$. Is it true that, for...
How large can a union-free collection $\mathcal{F}$ of subsets of $[n]$ be? By union-free we mean there are no solutions to $A\cup B=C$ with distinct...
Let $\delta(n)$ denote the density of integers which are divisible by some integer in $(n,2n)$. What is the growth rate of $\delta(n)$?If...
Is it true that, for any $c>1/2$, if $p$ is a sufficiently large prime then, for any $n\geq 0$, there exist $a,b\in(n,n+p^c)$ such that $ab\equiv...
Let $A\subseteq\mathbb{N}$ be infinite and $d_A(n)$ count the number of $a\in A$ which divide $n$. Is it true that, for every $k$,\[\limsup_{x\to...
Let $m,n\geq 1$. What is\[\# \{ k(m-k) : 1\leq k\leq m/2\} \cap \{ l(n-l) : 1\leq l\leq n/2\}?\]Can it be arbitrarily large? Is it $\leq (mn)^{o(1)}$...
Is it true that if $A\subseteq\mathbb{N}$ is such that\[\frac{1}{\log\log x}\sum_{n\in A\cap [1,x)}\frac{1}{n}\to \infty\]then\[\left(\sum_{n\in...
Let $N\geq 1$. What is the size of the largest $A\subset \{1,\ldots,N\}$ such that $[a,b]\leq N$ for all $a,b\in A$, where $[a,b]$ is the least...
Let $A=\{a_1
Is it true that, in any finite colouring of the integers, there must be two integers $x\neq y$ of the same colour such that $x+y$ is a square? What...
How large can $A\subseteq \{1,\ldots,N\}$ be if $A+A$ contains no square numbers?
Let $1\leq a_1<\cdots
If $p$ is a prime and $k,m\geq 2$ then let $r(k,m,p)$ be the minimal $r$ such that $r,r+1,\ldots,r+m-1$ are all $k$th power residues modulo $p$....
Let $n\in\mathbb{N}$ with $n\neq p^k$ for any prime $p$ and $k\geq 0$. What is the largest integer not of the form\[\sum_{1\leq...
Let $k\leq n$. What choice of $A\subseteq \{1,\ldots,n\}$ (with $\mathrm{gcd}(A)=1$) of size $\lvert A\rvert=k$ maximises the number of integers not...
If $A\subset \mathbb{N}$ is a finite set then let $G(A)$ denote the greatest integer which is not expressible as a finite sum of elements from $A$...
Let $A,B\subseteq \mathbb{N}$ be two infinite sets. How dense can $A+B$ be if all elements of $A+B$ are pairwise relatively prime?
Are there two infinite sets $A$ and $B$ such that $A+B$ agrees with the set of prime numbers up to finitely many exceptions?
Fix some integer $n$ and define a decreasing sequence in $[1,n)$ by $a_1=n-1$ and, for $k\geq 2$, letting $a_k$ be the greatest integer in...
Is it true that, if $A\subseteq \mathbb{N}$ is sparse enough and does not cover all residue classes modulo $p$ for any prime $p$, then there exists...
Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$, all of $n-a$ are prime for all $a\in A$ with $0
Is it true that, for every $n$ and $d$, there exists $k$ such that\[d \mid p_{n+1}+\cdots+p_{n+k},\]where $p_r$ denotes the $r$th prime?
We say $H$ is a unique subgraph of $G$ if there is exactly one way to find $H$ as a subgraph (not necessarily induced) of $G$. Is there a graph on...
Let $F(n)$ be the maximum possible size of a subset $A\subseteq\{1,\ldots,N\}$ such that the products $ab$ are distinct for all $a
Let $a_1=2$ and $a_2=3$ and continue the sequence by appending to $a_1,\ldots,a_n$ all possible values of $a_ia_j-1$ with $i\neq j$. Is it true that...
Let $a_1=1$ and $a_2=2$ and for $k\geq 3$ we choose $a_k$ to be the least integer $>a_{k-1}$ which is the sum of at least two consecutive terms of...
Let $f(1)=f(2)=1$ and for $n>2$\[f(n) = f(n-f(n-1))+f(n-f(n-2)).\]Does $f(n)$ miss infinitely many integers? What is its behaviour?
Is there a sequence $1\leq d_1
If $\tau(n)$ counts the number of divisors of $n$ then let\[F(f,n)=\frac{\tau((n+\lfloor f(n)\rfloor)!)}{\tau(n!)}.\]Is it true that\[\lim_{n\to...
If $\tau(n)$ counts the number of divisors of $n$, then what is the set of limit points of\[\frac{\tau((n+1)!)}{\tau(n!)}?\]
Are there infinitely many positive integers not of the form $n-\phi(n)$?
Let\[V'(x)=\#\{\phi(m) : 1\leq m\leq x\}\]and\[V(x)=\#\{\phi(m) \leq x : 1\leq m\}.\]Does $\lim V(x)/V'(x)$ exist? Is it $>1$?
Let $V(x)$ count the number of $n\leq x$ such that $\phi(m)=n$ is solvable. Does $V(2x)/V(x)\to 2$? Is there an asymptotic formula for $V(x)$?
For any $n$ let $F(n)$ be the largest $k$ such that any of the $k!$ possible ordering patterns appears in some sequence of...
Let $h_1(n)=h(n)=n+\tau(n)$ (where $\tau(n)$ counts the number of divisors of $n$) and $h_k(n)=h(h_{k-1}(n))$. Is it true, for any $m,n$, there exist...
Let $\omega(n)$ count the number of distinct primes dividing $n$. Are there infinitely many $n$ such that, for all $m
Let $\sigma_1(n)=\sigma(n)$, the sum of divisors function, and $\sigma_k(n)=\sigma(\sigma_{k-1}(n))$.Is it true that, for every $m,n\geq 2$, there...
Let $g_1=g(n)=n+\phi(n)$ and $g_k(n)=g(g_{k-1}(n))$. For which $n$ and $r$ is it true that $g_{k+r}(n)=2g_k(n)$ for all large $k$?
Let $\sigma_1(n)=\sigma(n)$, the sum of divisors function, and $\sigma_k(n)=\sigma(\sigma_{k-1}(n))$. Is it true that\[\lim_{k\to \infty}...
How many iterations of $n\mapsto \phi(n)+1$ are needed before a prime is reached? Can infinitely many $n$ reach the same prime? What is the density...
Let $\phi(n)$ be the Euler totient function and $\phi_k(n)$ be the iterated $\phi$ function, so that $\phi_1(n)=\phi(n)$ and...
Let $w(n)$ count the number of solutions to\[n=2^a+3^b+2^c3^d\]with $a,b,c,d\geq 0$ integers. Is it true that $w(n)$ is bounded by some absolute...
Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$?
Let $p$ be an odd prime. Is it true that the equation\[(p-1)!+a^{p-1}=p^k\]has only finitely many solutions?
For which integers $a\geq 1$ and primes $p$ is there a finite upper bound on those $k$ such that there are $a=a_1<\cdots
Does the equation\[2^m=a_1!+\cdots+a_k!\]with $a_1
Prove that, for any finite set $A\subset\mathbb{N}$, there exist $a,b\in A$ such that\[\mathrm{gcd}(a,b)\leq a/\lvert A\rvert.\]
Is there some function $f(r)$ such that $f(r)\to \infty$ as $r\to\infty$, such that, for infinitely many $n$, there exist $a_1,a_2$ with\[a_1+a_2>...
For any $k\geq 2$ let $g_k(n)$ denote the maximum value of\[(a_1+\cdots+a_k)-n\]where $a_1,\ldots,a_k$ are integers such that $a_1!\cdots a_k! \mid...
Is it true that there are no solutions to\[n! = x^k\pm y^k\]with $x,y,n\in \mathbb{N}$, with $xy>1$ and $k>2$?
Are the only solutions to\[n!=x^2-1\]when $n=4,5,7$?
Are there only finitely many solutions to\[\prod_i \binom{2m_i}{m_i}=\prod_j \binom{2n_j}{n_j}\]with the $m_i,n_j$ distinct?
Is it true that for every $k$ there exists $n$ such that\[\prod_{0\leq i\leq k}(n-i) \mid \binom{2n}{n}?\]
If $z_1,\ldots,z_n\in \mathbb{C}$ with $\lvert z_i\rvert=1$ then is it true that the probability that\[\lvert...
Let $t_k(n)$ denote the least $m$ such that\[n\mid m(m+1)(m+2)\cdots (m+k-1).\]Is it true that\[\sum_{n\leq x}t_2(n)\ll \frac{x^2}{(\log x)^c}\]for...
Let $f(n)$ denote the minimal $m\geq 1$ such that\[n! = a_1\cdots a_t\]with $a_1<\cdots
Let $A(n)$ denote the least value of $t$ such that\[n!=a_1\cdots a_t\]with $a_1\leq \cdots \leq a_t\leq n^2$. Is it true...
Let $t(n)$ be maximal such that there is a representation\[n!=a_1\cdots a_n\]with $t(n)=a_1\leq \cdots \leq a_n$. Obtain good bounds for $t(n)/n$. In...
Let $f(n)$ be the minimal $m$ such that\[n! = a_1\cdots a_k\]with $n< a_1<\cdots
Is it true that for every $n\geq 1$ there is a $k$ such that\[n(n+1)\cdots(n+k-1)\mid (n+k)\cdots (n+2k-1)?\]
Can one classify all solutions of\[\prod_{1\leq i\leq k_1}(m_1+i)=\prod_{1\leq j\leq k_2}(m_2+j)\]where $k_1,k_2>3$ and $m_1+k_1\leq m_2$? Are there...
Is there an absolute constant $c>0$ such that, for all $1\leq k< n$, the binomial coefficient $\binom{n}{k}$ has a divisor in $(cn,n]$?
Let $2\leq k\leq n-2$. Can $\binom{n}{k}$ be the product of consecutive primes infinitely often? For example\[\binom{21}{2}=2\cdot 3\cdot 5\cdot 7.\]
Let\[F(n) = \max_{\substack{m
If $1
Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime divisor of\[\prod_{0\leq i\leq k}(p^2+i)\]is $p$?
Let $u\leq v$ be such that the largest prime dividing $\prod_{u\leq m\leq v}m$ appears with exponent at least $2$. Is it true that $v-u=v^{o(1)}$?...
A number $n$ ishighly compositeif $\tau(m)<\tau(n)$ for all $m
We call an interval $[u,v]$ 'bad' if the greatest prime factor of $\prod_{u\leq m\leq v}m$ occurs with an exponent greater than $1$. Let $B(x)$ count...
Let $S(n)$ denote the largest integer such that, for all $1\leq k
Let $r\geq 0$. Does the density of integers $n$ for which $\binom{n}{k}$ is squarefree for at least $r$ values of $1\leq k
Is there some absolute constant $C>0$ such that\[\sum_{p\leq n}1_{p\nmid \binom{2n}{n}}\frac{1}{p}\leq C\]for all $n$ (where the summation is...
Are there infinitely many $n$ such that $\binom{2n}{n}$ is coprime to $105$?
Is it true that for any $n,k\geq 1$, if $n+1,\ldots,n+k$ are all composite then there are distinct primes $p_1,\ldots,p_k$ such that $p_i\mid n+i$...
For any $m\in \mathbb{N}$, let $F(m)$ be the minimal $k\geq 2$ (if it exists) such that there are $a_1<\cdots
Show that the equation\[n! = a_1!a_2!\cdots a_k!,\]with $n-1>a_1\geq a_2\geq \cdots \geq a_k\geq 2$, has only finitely many solutions.
Let $P(n)$ denote the largest prime factor of $n$. There are infinitely many $n$ such that $P(n)>P(n+1)>P(n+2)$.
Let $P(n)$ denote the largest prime factor of $n$. Show that the set of $n$ with $P(n)
Are there infinitely many $n$ such that the largest prime factor of $n$ is $
Let $\epsilon>0$ and $k\geq 2$. Is it true that, for all sufficiently large $n$, there is a sequence of $k$ consecutive integers in $\{1,\ldots,n\}$...
How large is the largest prime factor of $n(n+1)$?
Let $B_2(n)$ be the 2-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all primes that divide $n$ exactly once). Is it true...
Are there any 2-full $n$ such that $n+1$ is 3-full? That is, if $p\mid n$ then $p^2\mid n$ and if $p\mid n+1$ then $p^3\mid n+1$.
Do all pairs of consecutivepowerful numbers$n$ and $n+1$ come from solutions toPell equations? In other words, must either $n$ or $n+1$ be a...
Are there any triples of consecutive positive integers all of which are powerful (i.e. if $p\mid n$ then $p^2\mid n$)?
Is it true that there are only finitely many collections of disjoint intervals $I_1,\ldots,I_n$ of size $\lvert I_i\rvert \geq 4$ for $1\leq i\leq n$...
Let $A\subseteq \mathbb{N}$ be a finite set of size $N$. Is it true that, for any fixed $t$, there are\[\ll \frac{2^N}{N^{3/2}}\]many $S\subseteq A$...
Let $c>0$ and $n$ be some large integer. What is the size of the largest $A\subseteq \{1,\ldots,\lfloor cn\rfloor\}$ such that $n$ is not a sum of a...
Let $f(n)$ be minimal such that $\{1,\ldots,n-1\}$ can be partitioned into $f(n)$ classes so that $n$ cannot be expressed as a sum of distinct...
Let $a_1
Let $A=\{a_1<\cdots\}$ be an infinite sequence of integers. Let $f(n)$ count the number of solutions to\[n=\sum_{u\leq i\leq v}a_i.\]Is there such an...
Let $1\leq a_1<\cdots
Is there some $c>0$ such that, for all sufficiently large $n$, there exist integers $a_1<\cdots
Is there a lacunary sequence $A\subseteq \mathbb{N}$ (so that $A=\{a_1
Let $\alpha,\beta\in \mathbb{R}_{>0}$ such that $\alpha/\beta$ is irrational. Is the multiset\[\{ \lfloor \alpha\rfloor,\lfloor...
Let $A\subseteq \mathbb{R}^2$ be a measurable set with infinite measure. Must $A$ contain the vertices of an isosceles trapezoid of area $1$? What...
Is there some $c>0$ such that every measurable $A\subseteq \mathbb{R}^2$ of measure $\geq c$ contains the vertices of a triangle of area 1?
Let $p(x)\in \mathbb{Q}[x]$. Is it true that\[A=\{ p(n)+1/n : n\in \mathbb{N}\}\]is strongly complete, in the sense that, for any finite set...
If $A\subset\mathbb{N}$ is a finite set of integers which is dissociated (that is, all of the subset sums are distinct) then\[\sum_{n\in...
For what values of $t,\alpha \in (0,\infty)$ is the sequence $\lfloor t\alpha^n\rfloor$ complete (that is, all sufficiently large integers are the...
For what values of $0\leq m
Is there a sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers with\[\lim \frac{a_{n+1}}{a_n}=2\]such that\[P(A')= \left\{\sum_{n\in B}n : B\subseteq...
Let $A=\{1\leq a_1< a_2<\cdots\}$ be a set of integers such that$A\backslash B$ is complete for any finite subset $B$ and$A\backslash B$ is not...
Let $A\subseteq \mathbb{N}$ be a complete sequence, and define the threshold of completeness $T(A)$ to be the least integer $m$ such that all $n\geq...
If $A\subseteq \mathbb{N}$ is a set of integers such that\[\lvert A\cap \{1,\ldots,N\}\rvert\gg N^{1/2}\]for all $N$ then must $A$ be subcomplete?...
If $A\subseteq \mathbb{N}$ is a multiset of integers such that\[\lvert A\cap \{1,\ldots,N\}\rvert\gg N\]for all $N$ then must $A$ be subcomplete?...
With $a_1=1$ and $a_2=2$ let $a_{n+1}$ for $n\geq 2$ be the least integer $>a_n$ which can be expressed uniquely as $a_i+a_j$ for $i
Let $A=\{a_1<\cdots
Let $A=\{1,2,4,8,13,21,31,45,66,81,97,\ldots\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer...
Let $A\subseteq \mathbb{N}$ be a basis of order $r$. Must the set of integers representable as the sum of exactly $r$distinctelements from $A$ have...
The restricted order of a basis is the least integer $t$ (if it exists) such that every large integer is the sum of at most $t$ distinct summands...
Let $A\subseteq \mathbb{N}$ be an additive basis (of any finite order) such that $\lvert A\cap \{1,\ldots,N\}\rvert=o(N)$. Is it true...
For $r\geq 2$ let $h(r)$ be the maximal finite $k$ such that there exists a basis $A\subseteq \mathbb{N}$ of order $r$ (so every large integer is the...
Let $d(A)$ denote the density of $A\subseteq \mathbb{N}$. Characterise those $A,B\subseteq \mathbb{N}$ with positive density such...
Find the best function $f(n)$ such that every $n$ can be written as $n=a+b$ where both $a,b$ are $f(n)$-smooth (that is, are not divisible by any...
Let $A\subseteq \mathbb{N}$ be a set of density zero. Does there exist a $B$ such that $A\subseteq B+B$ and\[\lvert B\cap \{1,\ldots,N\}\rvert...
Let $A\subseteq \mathbb{N}$ and $D(A)$ be the set of those numbers which occur infinitely often as $a_1-a_2$ with $a_1,a_2\in A$. What conditions on...
Let $A,B\subseteq \mathbb{N}$ such that for all large $N$\[\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2}\]and\[\lvert B\cap \{1,\ldots,N\}\rvert \gg...
Does there exist a minimal basis with positive density, say $A\subset\mathbb{N}$, such that for any $n\in A$ the (upper) density of integers which...
Suppose $A\subseteq \mathbb{N}$ is a Sidon set. How large can\[\limsup_{N\to \infty}\frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}\]be?
Suppose $A\subseteq\mathbb{N}$ and $C>0$ is such that $1_A\ast 1_A(n)\leq C$ for all $n\in\mathbb{N}$. Can $A$ be partitioned into $t$ many subsets...
Suppose $A\subseteq \{1,\ldots,N\}$ is such that if $a,b\in A$ and $a\neq b$ then $a+b\nmid ab$. Can $A$ be 'substantially more' than the odd...
Let $A\subset \mathbb{N}$ be an additive basis of order $2$. Must there exist $B=\{b_1
Let $k\geq 3$ and $f_{k,3}(x)$ denote the number of integers $\leq x$ which are the sum of three nonnegative $k$th powers. Is it true...
Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a
Let $1\leq m\leq k$ and $f_{k,m}(x)$ denote the number of integers $\leq x$ which are the sum of $m$ many nonnegative $k$th powers. Is it true...
Let $k\geq 3$ and $A\subset \mathbb{N}$ be the set of $k$th powers. What is the order of growth of $1_A^{(k)}(n)$, i.e. the number of representations...
What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that all sums $\sum_{n\in S}\frac{1}{n}$ are distinct for $S\subseteq A$?
Let $S(N)$ count the number of distinct sums of the form $\sum_{n\in A}\frac{1}{n}$ for $A\subseteq \{1,\ldots,N\}$. Estimate $S(N)$.
What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there is a function $\delta:A\to \{-1,1\}$ such that\[\sum_{n\in...
Let $A\subseteq \mathbb{N}$ be an infinite arithmetic progression and $f:A\to \{-1,1\}$ be a non-constant function. Must there exist a finite...
Is there some constant $c>0$ such that for every $n\geq 1$ there exists some $\delta_k\in \{-1,0,1\}$ for $1\leq k\leq n$ with\[0< \left\lvert...
Is it true that if $A\subset \mathbb{N}\backslash\{1\}$ is a finite set with $\sum_{n\in A}\frac{1}{n}<2$ then there is a partition $A=A_1\sqcup A_2$...
Let $u_1=2$ and $u_{n+1}=u_n^2-u_n+1$. Let $a_1
Let $n\geq 1$ and let $m$ be minimal such that $\sum_{n\leq k\leq m}\frac{1}{k}\geq 1$. We define\[\epsilon(n) = \sum_{n\leq k\leq...
Are there infinitely many solutions to\[\frac{1}{p_1}+\cdots+\frac{1}{p_k}=1-\frac{1}{m},\]where $m\geq 2$ is an integer and $p_1<\cdots
Does there exist some $c>0$ such that, for any $K>1$, whenever $A$ is a sufficiently large finite multiset of integers with $\sum_{n\in...
What is the minimal value of $\lvert 1-\sum_{n\in A}\frac{1}{n}\rvert$ as $A$ ranges over all subsets of $\{1,\ldots,N\}$ which contain no $S$ such...
For integers $1\leq a
Let $\alpha >0$ and $N\geq 1$. Is it true that for any $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert \geq \alpha N$ there exists some $S\subseteq...
Let $N\geq 1$. How many integers can be written as the sum of distinct unit fractions with denominators from $\{1,\ldots,N\}$? Are there $o(\log N)$...
Let $N\geq 1$. What is the smallest integer not representable as the sum of distinct unit fractions with denominators from $\{1,\ldots,N\}$? Is it...
Are there two finite sets of primes $P,Q$ such that\[1=\left(\sum_{p\in P}\frac{1}{p}\right)\left(\sum_{q\in Q}\frac{1}{q}\right)?\]
Let $a/b\in \mathbb{Q}_{>0}$ with $b$ squarefree. Are there integers $1
For integers $1\leq a
Is it true that in any finite colouring of the integers there exists a monochromatic solution to\[\frac{1}{a}=\frac{1}{b}+\frac{1}{c}\]with distinct...
Let $f(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there are no solutions to\[\frac{1}{a}= \frac{1}{b}+\frac{1}{c}\]with...
Let $f(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there are no solutions to\[\frac{1}{a}\neq...
Let $A(N)$ denote the maximal cardinality of $A\subseteq \{1,\ldots,N\}$ such that $\sum_{n\in S}\frac{1}{n}\neq 1$ for all $S\subseteq A$. Estimate...
Is there an infinite sequence $a_1
Does every set $A\subseteq \mathbb{N}$ of positive density contain some finite $S\subset A$ such that $\sum_{n\in S}\frac{1}{n}=1$?
Let $N\geq 1$. How many $A\subseteq \{1,\ldots,N\}$ are there such that $\sum_{n\in A}\frac{1}{n}=1$?
Let $N\geq 1$ and let $k(N)$ be maximal such that there are $k$ disjoint $A_1,\ldots,A_k\subseteq \{1,\ldots,N\}$ with $\sum_{n\in A_i}\frac{1}{n}=1$...
Let $N\geq 1$ and let $k(N)$ denote the smallest $k$ such that there exist $N\leq n_1<\cdots
Let $N\geq 1$ and let $t(N)$ be the least integer $t$ such that there is no solution to\[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}\]with $t=n_1<\cdots...
Let $k\geq 1$ and let $v(k)$ be the minimal integer which does not appear as some $n_i$ in a solution to\[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}\]with...
Let $A$ be the set of $n\in \mathbb{N}$ such that there exist $1\leq m_1<\cdots
Let $n\geq 1$ and define $L_n$ to be the least common multiple of $\{1,\ldots,n\}$ and $a_n$ by\[\sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}.\]Is...
Let $a\geq 1$. Must there exist some $b>a$ such that\[\sum_{a\leq n\leq b}\frac{1}{n}=\frac{r_1}{s_1}\textrm{ and }\sum_{a\leq n\leq...
Is it true that, for all sufficiently large $k$, there exist finite intervals $I_1,\ldots,I_k\subset \mathbb{N}$, distinct, not overlapping or...
Is it true that there are only finitely many pairs of intervals $I_1,I_2$ such that\[\sum_{n_1\in I_1}\frac{1}{n_1}+\sum_{n_2\in I_2}\frac{1}{n_2}\in...
Let $k\geq 2$. Is it true that, for any distinct integers $1
Let $k\geq 2$. Is it true that there exists an interval $I$ of width $(e-1+o(1))k$ and integers $n_1<\cdots
Let $f(k)$ be the minimal value of $n_k$ such that there exist $n_1
Let $f(k)$ be the maximal value of $n_1$ such that there exist $n_1
Let $p:\mathbb{Z}\to \mathbb{Z}$ be a polynomial whose leading coefficient is positive and such that there exists no $d\geq 2$ with $d\mid p(n)$ for...
Let $A\subseteq \mathbb{N}$ be an infinite set and consider the following greedy algorithm for a rational $x\in (0,1)$: choose the minimal $n\in A$...
Let $n_1
Let $n_1
Let $k\geq 3$. Is there a choice of congruence classes $a_p\pmod{p}$ for every prime $p$ such that all except finitely many integers can be written...
Let $A=\{n_1<\cdots
Is it true that, for every $c$, there exists an $n$ such that $\sigma(n)>cn$ but there is no covering system whose moduli all divide $n$?
Is there an infinite Lucas sequence $a_0,a_1,\ldots$ where $a_{n+2}=a_{n+1}+a_n$ for $n\geq 0$ such that all $a_k$ are composite, and yet no integer...
If a finite system of $r$ congruences $\{ a_i\pmod{n_i} : 1\leq i\leq r\}$ (the $n_i$ are not necessarily distinct) covers $2^r$ consecutive integers...
If $G$ is a group then can there exist an exact covering of $G$ by more than one cosets of different sizes? (i.e. each element is contained in...
Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p\geq 5$?
Let $N\geq 1$. What is the largest $t$ such that there are $A_1,\ldots,A_t\subseteq \{1,\ldots,N\}$ with $A_i\cap A_j$ a non-empty arithmetic...
For any $n$, let $A(n)=\{0
Let $f(n)\to \infty$ as $n\to \infty$. Is it true that\[\sum_{n\geq 1} \frac{1}{(n+1)\cdots (n+f(n))}\]is irrational?
Let $P$ be a finite set of primes with $\lvert P\rvert \geq 2$ and let $\{a_1
Let $X\subseteq \mathbb{R}^3$ be the set of all points of the shape\[\left( \sum_{n\in A} \frac{1}{n},\sum_{n\in A}\frac{1}{n+1},\sum_{n\in A}...
Let $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ be the Fibonacci sequence. Let $n_1
Let $a_n$ be an infinite sequence of positive integers such that $\sum \frac{1}{a_n}$ converges. There exists some integer $t\geq 1$ such that\[\sum...
How fast can $a_n\to \infty$ grow if\[\sum\frac{1}{a_n}\quad\textrm{and}\quad\sum\frac{1}{a_n-1}\]are both rational?
Let $a_n$ be a sequence of integers such that for every bounded sequence of integers $b_n$ (with $a_n+b_n\neq 0$ and $b_n\neq 0$ for all $n$) the...
Let $a_n$ be a sequence of integers such that for every sequence of integers $b_n$ with $b_n/a_n\to 1$ the sum\[\sum\frac{1}{b_n}\]is irrational. Is...
Suppose $a_1
Are there infinitely many $n$ such that there exists some $t\geq 2$ and distinct integers $a_1,\ldots,a_t\geq 1$ such that\[\frac{n}{2^n}=\sum_{1\leq...
Let $a_1
Is the sum\[\sum_{n} \mu(n)^2\frac{n}{2^n}\]irrational?
Let $a_1,a_2,\ldots$ be a sequence of integers with $a_n\to \infty$. Is\[\sum_{n} \frac{\tau(n)}{a_1\cdots a_n}\]irrational, where $\tau(n)$ is the...
Let $A\subseteq \mathbb{N}$ be an infinite set. Is\[\sum_{n\in A}\frac{1}{2^n-1}\]irrational?
Let $n\geq 1$ and $f(n)$ be maximal such that for every $a_1\leq \cdots \leq a_n\in \mathbb{N}$ we have\[\max_{\lvert z\rvert=1}\left\lvert...
Let $z_1,z_2,\ldots \in [0,1]$ be an infinite sequence, and define the discrepancy\[D_N(I) = \#\{ n\leq N : z_n\in I\} - N\lvert I\rvert.\]Must there...
Let $A\subseteq \mathbb{N}$ be such that\[\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert \to \infty\textrm{ as }x\to \infty\]and\[\sum_{n\in A}...
Let $a_1
Let $k\geq 1$ and $\sigma_k(n)=\sum_{d\mid n}d^k$. Is\[\sum \frac{\sigma_k(n)}{n!}\]irrational?
Is\[\sum \frac{p_n}{2^n}\]irrational? (Here $p_n$ is the $n$th prime.)
Is\[\sum \frac{\sigma(n)}{2^n}\]irrational? (Here $\sigma(n)$ is the sum of divisors function.)
Is\[\sum_n \frac{\phi(n)}{2^n}\]irrational? Here $\phi$ is theEuler totient function.
Are there infinitely many $n$ such that, for all $k\geq 1$,\[ \omega(n+k) \ll k?\](Here $\omega(n)$ is the number of distinct prime divisors of $n$.)
Let $a_1
Let $(a,b)=1$. The set $\{a^kb^l: k,l\geq 0\}$ is complete - that is, every large integer is the sum of distinct integers of the form $a^kb^l$ with...
Let $A\subseteq \mathbb{N}$ be an infinite set such that $\lvert A\cap \{1,\ldots,N\}\rvert=o(N)$. Is it true that\[\limsup_{N\to \infty}\frac{\lvert...
Let $C>1$. Does the set of integers of the form $p+\lfloor C^k\rfloor$, for some prime $p$ and $k\geq 0$, have density $>0$?
Let $a_1
For every $n>2$ there exist distinct integers $1\leq x
Let $f(N)$ be the maximum size of $A\subseteq \{1,\ldots,N\}$ such that the sums $a+b+c$ with $a,b,c\in A$ are all distinct (aside from the trivial...
Is there an infinite set of primes $P$ such that if $\{a_1
Let $f:\mathbb{N}\to \{-1,1\}$ be a multiplicative function. Is it true that\[ \lim_{N\to \infty}\frac{1}{N}\sum_{n\leq N}f(n)\]always exists?
Let $c_1,c_2>0$. Is it true that, for any sufficiently large $x$, there exist more than $c_1\log x$ many consecutive primes $\leq x$ such that the...
Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap \{1,\ldots,N\}\rvert \gg \log N$ for all large $N$. Let $f(n)$ count the number of...
Let $f(n)$ count the number of solutions to $n=p+2^k$ for prime $p$ and $k\geq 0$. Is it true that $f(n)=o(\log n)$?
Let $N_k=2\cdot 3\cdots p_k$ and $\{a_1
For every $c\geq 0$ the density $f(c)$ of integers for which\[\frac{p_{n+1}-p_n}{\log n}< c\]exists and is a continuous function of $c$.
Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Prove that\[\sum_{1\leq n\leq N}d_n^2 \ll N(\log N)^2.\]
For $A\subset \mathbb{R}^2$ we define the upper density as\[\overline{\delta}(A)=\limsup_{R\to \infty}\frac{\lambda(A \cap...
Let $S$ be a string of length $2^k-1$ formed from an alphabet of $k$ characters. Must $S$ contain an abelian square: two consecutive blocks $x$ and...
Let $n\geq 1$ and\[A=\{a_1<\cdots
Are there arbitrarily long arithmetic progressions of primes?
Let $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\leq d_n$. Furthermore, there are...
For which $n$ are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, which determine $n-1$ distinct distances and so...
Let $g(k)$ be the smallest integer (if any such exists) such that any $g(k)$ points in $\mathbb{R}^2$ contains an empty convex $k$-gon (i.e. with no...
Does there exist $S\subseteq \mathbb{R}^2$ such that every set congruent to $S$ (that is, $S$ after some translation and rotation) contains exactly...
Let $S\subset \mathbb{R}^2$ be such that no two points in $S$ are distance $1$ apart. Must the complement of $S$ contain four points which form a...
Let $n\geq 4$. Are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, such that all pairwise distances are integers?
Is there a dense subset of $\mathbb{R}^2$ such that all pairwise distances are rational?
Let $1\leq k
Let $f(n)$ be minimal such that the following holds. For any $n$ points in $\mathbb{R}^2$, not all on a line, there must be at least $f(n)$ many...
Let $A$ be a finite collection of $d\geq 4$ non-parallel lines in $\mathbb{R}^2$ such that there are no points where at least four lines from $A$...
Let $s_1
For any $g\geq 2$, if $n$ is sufficiently large and $\equiv 1,3\pmod{6}$ then there exists a 3-uniform hypergraph on $n$ vertices such thatevery pair...
Let $x>0$ be a real number. For any $n\geq 1$ let\[R_n(x) = \sum_{i=1}^n\frac{1}{m_i}
Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\geq 0$, where $\Omega(m)<\log\log m$? (Here $\Omega(m)$ is the...
Are there $n$ such that there is a covering system with moduli the divisors of $n$ which is 'as disjoint as possible'?That is, for all $d\mid n$ with...
Is there an integer $m$ with $(m,6)=1$ such that none of $2^k3^\ell m+1$ are prime, for any $k,\ell\geq 0$?
Let $G_k(N)$ be such that any set of $N$ integers contains a subset of size at least $G_k(N)$ which does not contain a $k$-term arithmetic...
Does the longest arithmetic progression of primes in $\{1,\ldots,N\}$ have length $o(\log N)$?
If $A\subset \mathbb{R}$ does not contain a 3-term arithmetic progression then must $\mathbb{R}\backslash A$ contain an infinite arithmetic...
If $A\subset \mathbb{N}$ is a Sidon set then must the complement of $A$ contain an infinite arithmetic progression?
Can $\mathbb{N}$ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?
Must every permutation of $\mathbb{N}$ contain a monotone 4-term arithmetic progression? In other words, given a permutation $x$ of $\mathbb{N}$ must...
What is the largest $k$ such that in any permutation of $\mathbb{Z}$ there must exist a monotone $k$-term arithmetic progression $x_1<\cdots
Let $k\geq 3$. Must any ordering of $\mathbb{R}$ contain a monotone $k$-term arithmetic progression, that is, some $x_1<\cdots
Let $S\subseteq \mathbb{Z}^3$ be a finite set and let $A=\{a_1,a_2,\ldots,\}\subset \mathbb{Z}^3$ be an infinite $S$-walk, so that $a_{i+1}-a_i\in S$...
Let $A=\{a_1,a_2,\ldots\}\subset \mathbb{R}^d$ be an infinite sequence such that $a_{i+1}-a_i$ is a positive unit vector (i.e. is of the form...
Let $C>0$ be arbitrary. Is it true that, if $n$ is sufficiently large depending on $C$, then in any $2$-colouring of $\binom{\{2,\ldots,n\}}{2}$...
Let $H(k)$ be the smallest $N$ such that in any finite colouring of $\{1,\ldots,N\}$ (into any number of colours) there is always either a...
If $\mathbb{R}^2$ is finitely coloured then must there exist some colour class which contains the vertices of a rectangle of every area?
What is the smallest $k$ such that $\mathbb{R}^2$ can be red/blue coloured with no pair of red points unit distance apart, and no $k$-term arithmetic...
Find the best function $f(d)$ such that, in any 2-colouring of the integers, at least one colour class contains an arithmetic progression with common...
Let $F(N)$ be the maximal size of $A\subseteq \{1,\ldots,N\}$ which is 'non-averaging', so that no $n\in A$ is the arithmetic mean of at least two...
Let $f_3(n)$ be the maximal size of a subset of $\{0,1,2\}^n$ which contains no three points on a line. Is it true that $f_3(n)=o(3^n)$?
Any graph on $n$ vertices can be decomposed into $O(n)$ many edge-disjoint cycles and edges.
Let $R(3;k)$ be the minimal $n$ such that if the edges of $K_n$ are coloured with $k$ colours then there must exist a monochromatic triangle....
Let $k\geq 3$. What is the maximum number of edges that a graph on $n$ vertices can contain if it does not have a $k$-regular subgraph? Is it $\ll...
If $\mathcal{F}$ is a finite set of finite graphs then $\mathrm{ex}(n;\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have...
Let $1\leq k<\ell$ be integers and define $F_k(N,\ell)$ to be minimal such that every set $A\subset \mathbb{N}$ of size $N$ which contains at least...
Let $A_1,A_2,\ldots$ be an infinite collection of infinite sets of integers, say $A_i=\{a_{i1}
Find the smallest $h(d)$ such that the following holds. There exists a function $f:\mathbb{N}\to\{-1,1\}$ such that, for every $d\geq...
Let $N(k,\ell)$ be the minimal $N$ such that for any $f:\{1,\ldots,N\}\to\{-1,1\}$ there must exist a $k$-term arithmetic progression $P$ such that\[...
Show that, for any $n\geq 5$, the binomial coefficient $\binom{2n}{n}$ is not squarefree.
A finite set $A\subset \mathbb{R}^n$ is called Ramsey if, for any $k\geq 1$, there exists some $d=d(A,k)$ such that in any $k$-colouring of...
In any $2$-colouring of $\mathbb{R}^2$, for all but at most one triangle $T$, there is a monochromatic congruent copy of $T$.
Is it true that in any finite colouring of $\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements...
Is it true that for every $\epsilon>0$ and integer $t\geq 1$, if $N$ is sufficiently large and $A$ is a subset of $[t]^N$ of size at least $\epsilon...
Let $F(N)$ be the smallest possible size of $A\subset \{0,1,\ldots,N\}$ such that $\{0,1,\ldots,N\}\subset A-A$. Find the value of\[\lim_{N\to...
Let $k\geq 3$ and $f(k)$ be the supremum of $\sum_{n\in A}\frac{1}{n}$ as $A$ ranges over all sets of positive integers which do not contain a...
Let $F(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain any set of the form $\{n,2n,3n\}$. What is\[ \lim_{N\to...
If $G$ is a graph with at most $k$ edge disjoint triangles then can $G$ be made triangle-free after removing at most $2k$ edges?
Prove that\[R(4,k) \gg \frac{k^3}{(\log k)^{O(1)}}.\]
Give an asymptotic formula for $R(3,k)$.
A set $A\subset \mathbb{N}$ is primitive if no member of $A$ divides another. Is the sum\[\sum_{n\in A}\frac{1}{n\log n}\]maximised over all...
For any $d\geq 1$ if $H$ is a graph such that every subgraph contains a vertex of degree at most $d$ then $R(H)\ll_d n$.
Let $\alpha>0$ and $n\geq 1$. Let $F(n,\alpha)$ be the largest $k$ such that there exists some 2-colouring of the edges of $K_n$ in which any induced...
Let $\alpha\in[0,1/2)$ and $n,t\geq 1$. Let $F^{(t)}(n,\alpha)$ be the largest $m$ such that we can $2$-colour the edges of the complete $t$-uniform...
Let $h(N)$ be the smallest $k$ such that $\{1,\ldots,N\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must...
There exists some constant $c>0$ such that$$R(C_4,K_n) \ll n^{2-c}.$$
Let $A\subset \mathbb{N}$ be an infinite set such that, for any $n$, there are most $2$ solutions to $a+b=n$ with $a\leq b$....
Does there exist an infinite Sidon set which is an asymptotic basis of order 3?
Does there exist a maximal Sidon set $A\subset \{1,\ldots,N\}$ of size $O(N^{1/3})$?
Let $F(N)$ be the size of the largest Sidon subset of $\{1,\ldots,N\}$. Is it true that for every $k\geq 1$ we have\[F(N+k)\leq F(N)+1\]for all...
Let $A\subset \{1,\ldots,N\}$ be a Sidon set with $\lvert A\rvert\sim N^{1/2}$. Must $A+A$ be well-distributed over all small moduli? In particular,...
Let $A$ be a finite Sidon set and $A+A=\{s_1<\cdots
For any $M\geq 1$, if $A\subset \mathbb{N}$ is a sufficiently large finite Sidon set then there are at least $M$ many $a\in A+A$ such that...
For a graph $G$ let $\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ on at least two...
A minimal cut of a graph is a minimal set of vertices whose removal disconnects the graph. Let $c(n)$ be the maximum number of minimal cuts a graph...
Let $G$ be a graph with maximum degree $\Delta$. Is $G$ the union of at most $\tfrac{5}{4}\Delta^2$ sets of strongly independent edges (sets such...
Let $F(k)$ be the number of solutions to\[ 1= \frac{1}{n_1}+\cdots+\frac{1}{n_k},\]where $1\leq n_1<\cdots
If $H$ is bipartite with minimum degree $r$ then there exists $\epsilon=\epsilon(H)>0$ such that\[\mathrm{ex}(n;H) \gg n^{2-\frac{1}{r-1}+\epsilon}.\]
If $H$ is bipartite and is $r$-degenerate, that is, every induced subgraph of $H$ has minimum degree $\leq r$, then\[\mathrm{ex}(n;H) \ll n^{2-1/r}.\]
Let $s_1
The density of integers which have two divisors $d_1,d_2$ such that $d_1
Let $A\subset (1,\infty)$ be a countably infinite set such that for all $x\neq y\in A$ and integers $k\geq 1$ we have\[ \lvert kx -y\rvert \geq...
Let $r_3(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain a non-trivial $3$-term arithmetic progression. Prove that...
Let $r_k(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain a non-trivial $k$-term arithmetic progression. Prove that...
Let the van der Waerden number $W(k)$ be such that whenever $N\geq W(k)$ and $\{1,\ldots,N\}$ is $2$-coloured there must exist a monochromatic...
Let $k\geq 3$. Can the product of any $k$ consecutive integers $N$ ever be powerful? That is, must there always exist a prime $p\mid N$ such that...
Let $f(n)$ be the smallest number of colours required to colour the edges of $K_n$ such that every $K_4$ contains at least 5 colours. Determine the...
Let $A\subset \mathbb{R}^2$ be a set of $n$ points such that any subset of size $4$ determines at least $5$ distinct distances. Must $A$ determine...
Let $\epsilon,\delta>0$ and $n$ be sufficiently large in terms of $\epsilon$ and $\delta$. Let $G$ be a triangle-free graph on $n$ vertices with...
Let $f(n)$ be minimal such that every triangle-free graph $G$ with $n$ vertices and diameter $2$ contains a vertex with degree $\geq f(n)$.What is...
Let $A\subset \mathbb{R}^2$ be a set of $n$ points. Must there be two distances which occur at least once but between at most $n$ pairs of points?...
Let $F(N)$ be the maximal size of $A\subseteq\{1,\ldots,N\}$ such that no $a\in A$ divides the sum of any distinct elements of $A\backslash\{a\}$....
Let $A\subset\mathbb{R}^2$ be an infinite set which contains no three points on a line and no four points on a circle. Consider the graph with...
Let $R(n;k,r)$ be the smallest $N$ such that if the edges of $K_N$ are $r$-coloured then there is a set of $n$ vertices which does not contain a copy...
Let $G$ be a graph with $n$ vertices such that every induced subgraph on $\geq \lfloor n/2\rfloor$ vertices has more than $n^2/50$ edges. Must $G$...
Let $f(m)$ be maximal such that every graph with $m$ edges must contain a bipartite graph with\[\geq \frac{m}{2}+\frac{\sqrt{8m+1}-1}{8}+f(m)\]edges....
Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $\lvert A\rvert=n$ then $\prod_{a\neq b\in A}(a+b)$ has at least $f(n)$ distinct prime...
Let $A = \{ \sum\epsilon_k3^k : \epsilon_k\in \{0,1\}\}$ be the set of integers which have only the digits $0,1$ when written base $3$, and $B=\{...
For any $d\geq 1$ and $k\geq 0$ let $P(d,k)$ be the set of integers which are the sum of distinct powers $d^i$ with $i\geq k$. Let $3\leq...
Let $a,b,c$ be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form $a^kb^lc^m$ ($k,l,m\geq...
For which number theoretic functions $f$ is it true that, for any $F(n)$ such that $f(n)/F(n)\to 0$ for almost all $n$, there are infinitely many $x$...
Let $F_{k}(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that the product of no $k$ many distinct elements of $A$ is a square. Is...
Let $A\subseteq\mathbb{R}$ be an infinite set. Must there be a set $E\subset \mathbb{R}$ of positive measure which does not contain any set of the...
Let $z_i$ be an infinite sequence of complex numbers such that $\lvert z_i\rvert=1$ for all $i\geq 1$, and for $n\geq 1$ let\[p_n(z)=\prod_{i\leq n}...
Let $\alpha$ be a cardinal or ordinal number or an order type such that every two-colouring of $K_\alpha$ contains either a red $K_\alpha$ or a blue...
Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x\neq y$ such that $xy=yx$ can be...
Let $p(z)=\prod_{i=1}^n (z-z_i)$ for $\lvert z_i\rvert \leq 1$. Is it true that\[\lvert\{ z: \lvert p(z)\rvert <1\}\rvert>n^{-O(1)}\](or perhaps even...
If $p(z)$ is a polynomial of degree $n$ such that $\{z : \lvert p(z)\rvert\leq 1\}$ is connected then is it true...
If $p(z)\in\mathbb{C}[z]$ is a monic polynomial of degree $n$ then is the length of the curve $\{ z\in \mathbb{C} : \lvert p(z)\rvert=1\}$ maximised...
If $G$ is bipartite then $\mathrm{ex}(n;G)\ll n^{3/2}$ if and only $G$ is $2$-degenerate, that is, $G$ contains no induced subgraph with minimal...
Let $k=k(n,m)$ be minimal such that any directed graph on $k$ vertices must contain either an independent set of size $n$ or a transitive tournament...
If $G$ is a graph let $h_G(n)$ be defined such that any subgraph of $G$ on $n$ vertices can be made bipartite after deleting at most $h_G(n)$...
Is there some $F(n)$ such that every graph with chromatic number $\aleph_1$ has, for all large $n$, a subgraph with chromatic number $n$ on at most...
Any $A\subseteq \mathbb{N}$ of positive upper density contains a sumset $B+C$ where both $B$ and $C$ are infinite.
For every $r\geq 4$ and $k\geq 2$ is there some finite $f(k,r)$ such that every graph of chromatic number $\geq f(k,r)$ contains a subgraph of girth...
Let $f(n)$ be minimal such that any $f(n)$ points in $\mathbb{R}^2$, no three on a line, contain $n$ points which form the vertices of a convex...
Draw $n$ squares inside the unit square with no common interior point. Let $f(n)$ be the maximum possible sum of the side-lengths of the squares. Is...
Let $A,B\subset \mathbb{R}^2$ be disjoint sets of size $n$ and $n-3$ respectively, with not all of $A$ contained on a single line. Is there a line...
Given $n$ points in $\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$.
Let $h(n)$ count the number of incongruent sets of $n$ points in $\mathbb{R}^2$ which minimise the diameter subject to the constraint that...
Let $c>0$ and $h_c(n)$ be such that for any $n$ points in $\mathbb{R}^2$ such that there are $\geq cn^2$ lines each containing more than three...
Given $n$ points in $\mathbb{R}^2$, no five of which are on a line, the number of lines containing four points is $o(n^2)$.
Let $A$ be a set of $n$ points in $\mathbb{R}^2$ such that all pairwise distances are at least $1$ and if two distinct distances differ then they...
Let $A\subseteq\mathbb{R}^2$ be a set of $n$ points with minimum distance equal to 1, chosen to minimise the diameter of $A$. If $n$ is sufficiently...
Let $h(n)$ be such that any $n$ points in $\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ distinct...
Does every convex polygon have a vertex with no other $4$ vertices equidistant from it?
If $n$ points in $\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart.
Let $x_1,\ldots,x_n\in\mathbb{R}^2$ determine the set of distances $\{u_1,\ldots,u_t\}$. Suppose $u_i$ appears as the distance between $f(u_i)$ many...
Suppose $n$ points in $\mathbb{R}^2$ determine a convex polygon and the set of distances between them is $\{u_1,\ldots,u_t\}$. Suppose $u_i$ appears...
If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then they determine at least $\lfloor \frac{n}{2}\rfloor$ distinct distances.
Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\mathbb{R}^2$ in which every $x\in A$ has at least $f(n)$ points in $A$...
Suppose $A\subset \mathbb{R}^2$ has $\lvert A\rvert=n$ and minimises the number of distinct distances between points in $A$. Prove that for large $n$...
Does every set of $n$ distinct points in $\mathbb{R}^2$ contain at most $n^{1+O(1/\log\log n)}$ many pairs which are distance 1 apart?
Does every set of $n$ distinct points in $\mathbb{R}^2$ determine $\gg n/\sqrt{\log n}$ many distinct distances?
For any $\epsilon>0$ there exists $\delta=\delta(\epsilon)>0$ such that if $G$ is a graph on $n$ vertices with no independent set or clique of size...
Let $\epsilon >0$. Is it true that, if $k$ is sufficiently large, then\[R(G)>(1-\epsilon)^kR(k)\]for every graph $G$ with chromatic number...
Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that every subgraph of $Q_n$...
Let $n\geq 4$ and $f(n)$ be minimal such that every graph on $n$ vertices with minimal degree $\geq f(n)$ contains a $C_4$. Is it true that, for all...
The cycle set of a graph $G$ on $n$ vertices is a set $A\subseteq \{3,\ldots,n\}$ such that there is a cycle in $G$ of length $\ell$ if and only if...
Suppose that we have a family $\mathcal{F}$ of subsets of $[4n]$ such that $\lvert A\rvert=2n$ for all $A\in\mathcal{F}$ and for every $A,B\in...
Let $F(n)$ be maximal such that every graph on $n$ vertices contains a regular induced subgraph on at least $F(n)$ vertices. Prove that $F(n)/\log...
Let $G$ be a chordal graph on $n$ vertices - that is, $G$ has no induced cycles of length greater than $3$. Can the edges of $G$ be partitioned into...
Let $c>0$ and let $f_c(n)$ be the maximal $m$ such that every graph $G$ with $n$ vertices and at least $cn^2$ edges, where each edge is contained in...
We say $G$ is Ramsey size linear if $R(G,H)\ll m$ for all graphs $H$ with $m$ edges and no isolated vertices.Are there infinitely many graphs $G$...
Give a constructive proof that $R(k)>C^k$ for some constant $C>1$.
If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$,...
Is it true that in any $2$-colouring of the edges of $K_n$ there must exist at least\[(1+o(1))\frac{n^2}{12}\]many edge-disjoint monochromatic...
Is there a graph of chromatic number $\aleph_1$ such that for all $\epsilon>0$ if $n$ is sufficiently large and $H$ is a subgraph on $n$ vertices...
Let $f(n)\to \infty$ (possibly very slowly). Is there a graph of infinite chromatic number such that every finite subgraph on $n$ vertices can be...
Let $k\geq 0$. Let $G$ be a graph such that every subgraph $H$ contains an independent set of size $\geq (n-k)/2$, where $n$ is the number of...
Is there a set $A\subset \mathbb{N}$ of density $0$ and a constant $c>0$ such that every graph on sufficiently many vertices with average degree...
Is it true that for every infinite arithmetic progression $P$ which contains even numbers there is some constant $c=c(P)$ such that every graph with...
Let $\mathfrak{c}$ be the ordinal of the real numbers, $\beta$ be any countable ordinal, and $2\leq n<\omega$. Is it true that $\mathfrak{c}\to...
Is\[\sum_{n\geq 2}\frac{\omega(n)}{2^n}\]irrational? (Here $\omega(n)$ counts the number of distinct prime divisors of $n$.)
Is\[\sum_{n\geq 2}\frac{1}{n!-1}\]irrational?
If $f:\mathbb{N}\to \{-1,+1\}$ then is it true that for every $C>0$ there exist $d,m\geq 1$ such that\[\left\lvert \sum_{1\leq k\leq...
Is there $A\subseteq \mathbb{N}$ such that\[\lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n}\]exists and is $\neq 0$?
Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_1
Does every finite graph with minimum degree at least 3 contain a cycle of length $2^k$ for some $k\geq 2$?
Does every graph with infinite chromatic number contain a cycle of length $2^n$ for infinitely many $n$?
If $G_1,G_2$ are two graphs with chromatic number $\aleph_1$ then must there exist a graph $G$ whose chromatic number is $4$ (or even $\aleph_0$)...
For any graph $H$ is there some $c=c(H)>0$ such that every graph $G$ on $n$ vertices that does not contain $H$ as an induced subgraph contains either...
Does every graph on $n$ vertices with $>\mathrm{ex}(n;C_4)$ edges contain $\gg n^{1/2}$ many copies of $C_4$?
Is it true that the number of graphs on $n$ vertices which do not contain $G$ is\[\leq 2^{(1+o(1))\mathrm{ex}(n;G)}?\]
If $G$ is a graph which contains odd cycles of $\leq k$ different lengths then $\chi(G)\leq 2k+2$, with equality if and only if $G$ contains...
If $G$ is a graph with infinite chromatic number and $a_1
Let $N\geq p_k$ where $p_k$ is the $k$th prime. Suppose $A\subseteq \{1,\ldots,N\}$ is such that there are no $k+1$ elements of $A$ which are...
A set of integers $A$ is Ramsey $r$-complete if, whenever $A$ is $r$-coloured, all sufficiently large integers can be written as a monochromatic sum...
A set of integers $A$ is Ramsey $2$-complete if, whenever $A$ is $2$-coloured, all sufficiently large integers can be written as a monochromatic sum...
Let $A$ be a finite set of integers. Is it true that, for every $k$, if $\lvert A\rvert$ is sufficiently large depending on $k$, then there are least...
Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$\[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert...
Is there an infinite set $A\subset \mathbb{N}$ such that for every $a\in A$ there is an integer $n$ such that $\phi(n)=a$, and yet if $n_a$ is the...
Schoenberg proved that for every $c\in [0,1]$ the density of\[\{ n\in \mathbb{N} : \phi(n)
Let $A=\{a_1<\cdots
Are there infinitely many integers $n,m$ such that $\phi(n)=\sigma(m)$?
If $\delta>0$ and $N$ is sufficiently large in terms of $\delta$, and $A\subseteq\{1,\ldots,N\}$ is such that $\sum_{a\in A}\frac{1}{a}>\delta \log...
Does every finite colouring of the integers have a monochromatic solution to $1=\sum \frac{1}{n_i}$ with $2\leq n_1<\cdots
Let $k\geq 2$. Is there an integer $n_k$ such that, if $D=\{ 1
Let $N\geq 1$ and $A\subset \{1,\ldots,N\}$ be a Sidon set. Is it true that, for any $\epsilon>0$, there exist $M$ and $B\subset \{N+1,\ldots,M\}$...
If $A,B\subset \{1,\ldots,N\}$ are two Sidon sets such that $(A-A)\cap(B-B)=\{0\}$ then is it true that\[ \binom{\lvert A\rvert}{2}+\binom{\lvert...
Let $M\geq 1$ and $N$ be sufficiently large in terms of $M$. Is it true that for every Sidon set $A\subset \{1,\ldots,N\}$ there is another Sidon set...
Let $A\subset\mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c\in A$ (aside from the trivial...
For what functions $g(N)\to \infty$ is it true that\[\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}\]implies $\limsup 1_A\ast...
Is there an infinite Sidon set $A\subset \mathbb{N}$ such that\[\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}\]for all $\epsilon>0$?
Does there exist $B\subset\mathbb{N}$ which is not an additive basis, but is such that for every set $A\subseteq\mathbb{N}$ of Schnirelmann density...
We say that $A\subset \mathbb{N}$ is an essential component if $d_s(A+B)>d_s(B)$ for every $B\subset \mathbb{N}$ with $0
Find the optimal constant $c>0$ such that the following holds.For all sufficiently large $N$, if $A\sqcup B=\{1,\ldots,2N\}$ is a partition into two...
Let $B\subseteq\mathbb{N}$ be an additive basis of order $k$ with $0\in B$. Is it true that for every $A\subseteq\mathbb{N}$ we have\[d_s(A+B)\geq...
For any permutation $\pi\in S_n$ of $\{1,\ldots,n\}$ let $S(\pi)$ count the number of distinct consecutive sums, that is, sums of the shape...
Let $A\subset\mathbb{N}$ be such that every large integer can be written as $n^2+a$ for some $a\in A$ and $n\geq 0$. What is the smallest possible...
Is there a set $A\subset\mathbb{N}$ such that\[\lvert A\cap\{1,\ldots,N\}\rvert = o((\log N)^2)\]and such that every large integer can be written as...
Given any infinite set $A\subset \mathbb{N}$ there is a set $B$ of density $0$ such that $A+B$ contains all except finitely many integers.
Let $h(N)$ be the maximum size of a Sidon set in $\{1,\ldots,N\}$. Is it true that, for every $\epsilon>0$,\[h(N) = N^{1/2}+O_\epsilon(N^\epsilon)?\]
Is there an explicit construction of a set $A\subseteq \mathbb{N}$ such that $A+A=\mathbb{N}$ but $1_A\ast 1_A(n)=o(n^\epsilon)$ for every...
If $A\subseteq \mathbb{N}$ is such that $A+A$ contains all but finitely many integers then $\limsup 1_A\ast 1_A(n)=\infty$.
An $\epsilon$-almost covering system is a set of congruences $a_i\pmod{n_i}$ for distinct moduli $n_1<\ldots
Let $A\subset\mathbb{N}$ be infinite. Must there exist some $k\geq 1$ such that almost all integers have a divisor of the form $a+k$ for some $a\in...
Let $n_1
Does every triangle-free graph on $5n$ vertices contain at most $n^5$ copies of $C_5$?
Can every triangle-free graph on $5n$ vertices be made bipartite by deleting at most $n^2$ edges?
Let $\epsilon>0$ and $n$ be sufficiently large depending on $\epsilon$. Is there a graph on $n$ vertices with $\geq n^2/8$ many edges which contains...
Let $f(n)$ be minimal such that there is an intersecting family $\mathcal{F}$ of sets of size $n$ (so $A\cap B\neq\emptyset$ for all $A,B\in...
Erdos called this 'perhaps my first serious problem,' dating it to 1931.
A fundamental question connecting density and structure in number theory.
Asks whether the set of limit points of normalized prime gaps equals $[0,\infty)$.
Posed by Erdos and Selfridge. Selfridge offered $2000 for an explicit example.
Studies the representation of odd numbers as sums involving primes and powers of 2.
Erdos described this as 'probably unattackable.'
A classic question on representing odd numbers.
Studies the density of integers with unique representations as sums.
Erdos suggested computer exploration, seeing no other method to attack this.
Erdos characterized this conjecture as 'rather silly.'