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

Source 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.