Three‐colored asymmetric bipartite Ramsey number of connected matchings and cycles

Abstract

AbstractGuaranteed by Szemerédi's Regularity Lemma, a technique originated by Łuczak is to reduce the problem of showing the existence of a monochromatic cycle to show the existence of a monochromatic matching in a component. So determining the Ramsey number of connected matchings is crucial in determining the Ramsey number of cycles. Let be integers and be the minimum integer such that for any red‐blue‐green coloring of , there is a red matching of size at least in a component, or a blue matching of size at least in a component, or a green matching of size at least in a component. Bucić, Letzter, and Sudakov determined the exact value of and led to the asymptotic value of 3‐colored bipartite Ramsey number of cycles (symmetric case). In this paper, we determine the exact value of completely. This answers a question of Bucić, Letzter, and Sudakov. The crucial part of the proof is the construction we give in Section 4. Applying the technique of Łuczak, we obtain the asymptotic value of 3‐colored bipartite Ramsey number of cycles for all asymmetric cases.