Erdős Problem #483

declared status 'open'. Formalized: no. 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 $c>0$? Current best: 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]. [Ex94] Exoo, G., A lower bound for Schur numbers and multicolor Ramsey numbers. Prize: no. OEIS: A030126. Tags: additive combinatorics, number theory, ramsey theory.

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