Problem Statement
Let $R(3,3,n)$ denote the smallest integer $m$ such that if we $3$-colour the edges of $K_m$ then there is either a monochromatic triangle in one of the first two colours or a monochromatic $K_n$ in the third colour. Define $R(3,n)$ similarly but with two colours. Show that\[\frac{R(3,3,n)}{R(3,n)}\to \infty\]as $n\to \infty$.
Categories:
Graph Theory Ramsey Theory
Progress
A problem of Erdős and Sós. This was solved by Alon and Rödl [AlRo05], who in fact show that\[R(3,3,n)\asymp n^3(\log n)^{O(1)}\](recalling that Shearer [Sh83] showed $R(3,n) \ll n^2/\log n$).This problem is #22 in Ramsey Theory in the graphs problem collection. See also [925].
Source: erdosproblems.com/553 | Last verified: January 15, 2026