Ramsey numbers for bipartite graphs with small bandwidth

TL;DR

This paper provides asymptotic estimates for Ramsey numbers of bipartite graphs with small bandwidth and bounded degree, including grid graphs, for two and three colors, advancing understanding of their combinatorial properties.

Contribution

It determines asymptotically the two and three color Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree, including grid graphs.

Findings

Asymptotic formulas for two-color Ramsey numbers of grid graphs.

Asymptotic estimates for three-color Ramsey numbers under certain conditions.

Extension of known results to broader classes of bipartite graphs.

Abstract

We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine asymptotically the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for such graphs with the additional assumption that the bipartite graph is balanced.