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 $G$. Determine the value of \[R(C_{2n};k).\]
A problem of Erdős and Graham. Erdős [Er81c] gives the bounds \[k^{1+\frac{1}{2n}}\ll R(C_{2n};k)\ll k^{1+\frac{1}{n-1}}.\] Chung and Graham [ChGr75] showed that \[R(C_4;k)>k^2-k+1\] when $k-1$ is a prime power and \[R(C_4;k)\leq k^2+k+1\] for all $k$.

See also the entry in the graphs problem collection.