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.
