Problem 183

Erdős Problem #183: declared status 'open'. Formalized: no. 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. Determine$$\lim_{k\to \infty}R(3;k)^{1/k}.$$ Current best: The best-known upper bounds are all of the form $ck!+O(1)$, and arise from this type of inductive relationship and computational bounds for $R(3;k)$ for small $k$. The best-known lower bound (coming from lower bounds for Schur numbers) is$$R(3,k)\geq (380)^{k/5}-O(1),$$due to Ageron, Casteras, Pellerin, Portella, Rimmel, and Tomasik [ACPPRT21] (improving previous bounds of Exoo [Ex94] and Fredricksen and Sweet [FrSw00]). [Ex94] Exoo, G., A lower bound for Schur numbers and multicolor Ramsey numbers. Prize: $250. OEIS: A003323. Tags: graph theory, ramsey theory.