### Bipartite Ramsey Numbers for Graphs of Small Bandwidth

#### Abstract

A graph $H=(W,E_H)$ is said to have *bandwidth* at most $b$ if there exists a labeling of $W$ as $w_1,w_2,\dots,w_n$ such that $|i-j|\leq b$ for every edge $w_iw_j\in E_H$, and a *bipartite balanced $(\beta,\Delta)$-graph* $H$ is a bipartite graph with bandwidth at most $\beta |W|$ and maximum degree at most $\Delta$, and furthermore it has a proper 2-coloring $\chi :W\rightarrow[2]$ such that $||\chi^{-1}(1)|-|\chi^{-1}(2)||\leq\beta|\chi^{-1}(2)|$. We prove that for any fixed $0<\gamma<1$ and integer $\Delta\ge1$, there exist a constant $\beta=\beta(\gamma,\Delta)>0$ and a natural number $n_0$ such that for every balanced $(\beta,\Delta)$-graph $H$ on $n\geq n_0$ vertices the bipartite Ramsey number $br(H,H)$ is at most $(1+\gamma)n$. In particular, $br(C_{2n},C_{2n})=(2+o(1))n$.