All Random Solved Random Open
Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Show that \[\lim_{k\to \infty}\frac{R(C_{2n+1};k)}{R(K_3;k)}=0\] for any $n\geq 2$.
A problem of Erdős and Graham. The problem is open even for $n=2$.

See also the entry in the graphs problem collection.