Erdos Problems

Let $r_3(N)$ be the size of the largest subset of...

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...

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...

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...

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...

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...

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...

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...

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

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...

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...

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...

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...

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...

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 $A\subseteq\mathbb{N}$...

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...

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...

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

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 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...

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

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...

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...

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...

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...

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 :...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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 $n$...

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...

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 $n$...

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...

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...

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...

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...

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 $\mathbb{R}^2$,...

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 $4$...

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...

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...

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...

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...

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...

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...

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 $\mathbb{R}^2$...

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 $\mathbb{R}^2$...

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...

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 $\epsilon >0$

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...

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...

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...

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...

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...

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, $G$...

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...

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...

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...

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 $n$...

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 $K_n$...

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...

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)

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...

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...

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...

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,...

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?

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?

Is\[\sum_{n\geq 2}\frac{1}{n!-1}\]irrational?

If $f:\mathbb{N}\to \{-1,+1\}$ then is it true that for...

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...

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...

Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_1

Does every finite graph with minimum degree at least 3...

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...

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...

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...

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 $>\mathrm{ex}(n;C_4)$...

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...

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 $\leq k$...

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...

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

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 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 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

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

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...

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...

Schoenberg proved that for every $c\in [0,1]$ the density of\[\{ n\in \mathbb{N} : \phi(n)

Let $A=\{a_1<\cdots

Let $A=\{a_1<\cdots

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Are there infinitely many integers $n,m$ such that...

Are there infinitely many integers $n,m$ such that $\phi(n)=\sigma(m)$?

If $\delta>0$ and $N$ is sufficiently large in terms of...

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...

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...

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

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...

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 $M$

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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

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

Let $n_1

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Does every triangle-free graph on $5n$ vertices contain at...

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...

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...

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...

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...

Distinct Subset Sums

Erdos called this 'perhaps my first serious problem,' dating it to 1931.

Arithmetic Progressions in Dense Sets

A fundamental question connecting density and structure in number theory.

Limit Points of Prime Gaps

Asks whether the set of limit points of normalized prime gaps equals $[0,\infty)$.

Odd Covering Systems

Posed by Erdos and Selfridge. Selfridge offered $2000 for an explicit example.

Odd Numbers Not of Form p+2^k+2^l

Studies the representation of odd numbers as sums involving primes and powers of 2.

Prime Plus Powers of 2

Erdos described this as 'probably unattackable.'

Squarefree Plus Power of 2

A classic question on representing odd numbers.

Unique Representation Sums

Studies the density of integers with unique representations as sums.

Alternating Sum Over Primes

Erdos suggested computer exploration, seeing no other method to attack this.

Odd Numbers Not of Form 2^k+p

Erdos characterized this conjecture as 'rather silly.'