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arXiv · 2610.29813

The three-colour bipartite Ramsey number of sufficiently long even cycles

Jiangdong Ai·Yihan Yin
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Abstract

For a bipartite graph $H$, let $\operatorname{br}_3(H)$ be the least integer $N$ such that every three-colouring of $K_{N,N}$ contains a monochromatic copy of $H$. We prove that $\operatorname{br}_3(C_{2n})=3n$ for all sufficiently large $n$, and give the lower bound for every $n\ge2$. For the upper bound, we reduce a counterexample to a finite rational weighted profile. Cycle extension and vertex-cover arguments force the critical components to have six unique minimum covers which partition the profile. The remaining component supports give a contradiction.

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Jiangdong Ai, Yihan Yin. 2026-10-21. The three-colour bipartite Ramsey number of sufficiently long even cycles. https://arxiv.org/abs/2610.29813

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