Problem Statement
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 particular, is it true that $f(k) < c^k$ for some constant $c>0$?
Categories:
Number Theory Additive Combinatorics Ramsey Theory
Progress
The values of $f(k)$ are known as Schur numbers. The best-known bounds for large $k$ are\[(380)^{k/5}-O(1)\leq f(k) \leq \lfloor(e-\tfrac{1}{24}) k!\rfloor-1.\]The lower bound is due to Ageron, Casteras, Pellerin, Portella, Rimmel, and Tomasik [ACPPRT21] (improving previous bounds of Exoo [Ex94] and Fredricksen and Sweet [FrSw00]) and the upper bound is due to Whitehead [Wh73]. Note that $380^{1/5}\approx 3.2806$.The known values of $f$ are $f(1)=2$, $f(2)=5$, $f(3)=14$, $f(4)=45$, and $f(5)=161$ (see A030126). (The equality $f(5)=161$ was established by Heule [He17]).
See also [183] (in particular a folklore observation gives $f(k)\leq R(3;k)-1$).
Source: erdosproblems.com/483 | Last verified: January 15, 2026