Erdős Problem #183 — 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.

Erdős asked whether the exponential growth rate of the multicolour triangle Ramsey numbers is finite. The relevant limit is known to exist, but its finiteness or infinitude remains unresolved.

Known results

• Chung and Grinstead (1983): supermultiplicativity establishes existence of $\lim_{k\to\infty}R_k(3)^{1/k}$.
• Xu, Xie, and Chen (2002): $R_k(3)\leq (e-\tfrac16)k!+1$.
• Ageron, Casteras, Pellerin, Portella, Rimmel, and Tomasik (2021): $R_k(3)\geq 380^{k/5}-O(1)$, giving a rate at least $380^{1/5}\approx 3.2806$.
• Erdős favored the conjecture that the limit is infinite, but no proof is recorded.

Current status (as of September 2026): The limit and existing factorial upper and exponential lower bounds are settled, while whether the limit is finite or infinite remains open.