A transference principle for Ramsey numbers of bounded degree graphs

We investigate Ramsey numbers of bounded degree graphs and provide an interpolation between known results on the Ramsey numbers of general bounded degree graphs and bounded degree graphs of small bandwidth. Our main theorem implies that there exists a constant $c$ such that for every $\Delta$, there exists $\beta$ such that if $G$ is an $n$-vertex graph with maximum degree at most $\Delta$ having a homomorphism $f$ into a graph $H$ of maximum degree at most $d$ where $|f^{-1}(v)| \le \beta n$ for all $v \in V(H)$, then the Ramsey number of $G$ is at most $c^{d \log d} n$. A construction of Graham, R\"odl, and Ruci\'nski shows that the statement above holds only if $\beta \le (c')^{\Delta}$ for some constant $c' < 1$. We further study the parameter $\beta$ using a density-type embedding theorem for bipartite graphs of small bandwidth. This theorem may be of independent interest.