Locally seeded embeddings, and Ramsey numbers of bipartite graphs with sublinear bandwidth

A seminal result of Lee asserts that the Ramsey number of any bipartite $d$-degenerate graph $H$ satisfies $\log r(H) = \log n + O(d)$. In particular, this bound applies to every bipartite graph of maximal degree $Δ$. It remains a compelling challenge to identify conditions that guarantee that an $n$-vertex graph $H$ has Ramsey number linear in $n$, independently of $Δ$. Our contribution is a characterization of bipartite graphs with linear-size Ramsey numbers in terms of graph bandwidth, a notion of local connectivity. We prove that for any $n$-vertex bipartite graph $H$ with maximal degree at most $Δ$ and bandwidth $b(H)$ at most $\exp(-CΔ\logΔ)\,n$, we have $\log r(H) = \log n + O(1)$. This characterization is nearly optimal: for every $Δ$ there exists an $n$-vertex bipartite graph $H$ of degree at most $Δ$ and $b(H) \leq \exp(-cΔ)\,n$, such that $\log r(H) = \log n + Ω(Δ)$. We also provide bounds interpolating between these two bandwidth regimes.