The minimum number of colors in a (C_{2k},3)-coloring of K_{n,n} is exactly (7/20)n+o(n) for k=3 and lies between improved explicit bounds for all k≥4.
Generalized Ramsey numbers of cycles, paths, and hypergraphs
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abstract
Given a $k$-uniform hypergraph $G$ and a set of $k$-uniform hypergraphs $\mathcal{H}$, the generalized Ramsey number $f(G,\mathcal{H},q)$ is the minimum number of colors needed to edge-color $G$ so that every copy of every hypergraph $H\in \mathcal{H}$ in $G$ receives at least $q$ different colors. In this note we obtain bounds, some asymptotically sharp, on several generalized Ramsey numbers, when $G=K_n$ or $G=K_{n,n}$ and $\mathcal{H}$ is a set of cycles or paths, and when $G=K_n^k$ and $\mathcal{H}$ contains a clique on $k+2$ vertices or a tight cycle.
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Edge-coloring $K_{n, n}$ with no 2-colored $C_{2k}$
The minimum number of colors in a (C_{2k},3)-coloring of K_{n,n} is exactly (7/20)n+o(n) for k=3 and lies between improved explicit bounds for all k≥4.