Three-colour bipartite Ramsey number for graphs with small bandwidth

TL;DR

This paper estimates the 3-colour bipartite Ramsey number for certain small bandwidth graphs, providing asymptotic bounds and exact results for balanced grid graphs.

Contribution

It establishes an asymptotic upper bound for the 3-colour bipartite Ramsey number of graphs with small bandwidth and bounded degree, including balanced grid graphs.

Findings

Asymptotic bound of (3/2+o(1))|V(H)| for the Ramsey number

Exact asymptotics for balanced grid graphs

Extension of bipartite Ramsey theory to small bandwidth graphs

Abstract

We estimate the $3$-colour bipartite Ramsey number for balanced bipartite graphs $H$ with small bandwidth and bounded maximum degree. More precisely, we show that the minimum value of $N$ such that in any $3$-edge colouring of $K_{N,N}$ there is a monochromatic copy of $H$ is at most $\big(3/2+o(1)\big)|V(H)|$. In particular, we determine asymptotically the $3$-colour bipartite Ramsey number for balanced grid graphs.