Problem Statement
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\[R(C_{2n};k).\]
Categories:
Graph Theory Ramsey Theory
Progress
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 $k$.This problem is #24 in Ramsey Theory in the graphs problem collection.
Source: erdosproblems.com/555 | Last verified: January 15, 2026