Let $k$ be a fixed positive integer. All edges of the complete graph $K_n$ are colored in one of $k$ colors. What is the least $n$ such that there always exists a $4$-cycle of the same color?
This sounds like Ramsey, but the $4$-cycle doesn't fit well.
There is a decent amount of literature on calculating the diagonal Ramsey numbers $R_k(C_4)$ (where $R_k(G)$ is the smallest $n$ such that any $k$-coloring of the edges of $K_n$ contains a monocolored $G$). A recent survey by Radziszowski contains many useful references. In particular, the asymptotic behavior is known to be $R_k(C_4)\in\Theta(k^2)$; the upper bound $R_k(C_4)\le k^2 + k + 1$ holds for all $k\ge 1$, as does the lower bound $R_k(C_4) \ge 2k+1$; and the much tighter lower bound $R_k(C_4)\ge k^2+2$ holds when $k$ is a prime power. Finally, $R_4(C_4)=18$, and $R_5(C_4) \in \{27,28,29\}$.