The generalized Ramsey number r(G,H,q) is the minimum number of colors needed to color the edges of G such that every isomorphic copy of H has at least q colors. In this note, we improve the upper and lower bounds on r(Kn,n,C2k,3). Our upper bound answers a question of Lane and Morrison. For k=3 we obtain the asymptotically sharp estimate r(Kn,n,C6,3)=207n+o(n).
@article{arxiv.2507.13329,
title = {Edge-coloring $K_{n, n}$ with no 2-colored $C_{2k}$},
author = {Deepak Bal and Patrick Bennett},
journal= {arXiv preprint arXiv:2507.13329},
year = {2025}
}