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

TL;DR

This paper characterizes bipartite graphs with linear Ramsey numbers based on their bandwidth, showing that low bandwidth ensures linearity in the Ramsey number, and provides bounds for different bandwidth regimes.

Contribution

It introduces a nearly optimal bandwidth condition that guarantees linear Ramsey numbers for bipartite graphs, extending previous results and providing bounds for various bandwidth levels.

Findings

Graphs with bandwidth below a certain exponential threshold have linear Ramsey numbers.

Existence of bipartite graphs with higher bandwidth and superlinear Ramsey numbers.

Bounds that interpolate between different bandwidth regimes.

Abstract

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 $\Delta$. It remains a compelling challenge to identify conditions that guarantee that an $n$-vertex graph $H$ has Ramsey number linear in $n$, independently of $\Delta$. 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 $\Delta$ and bandwidth $b(H)$ at most $\exp(-C\Delta\log\Delta)\,n$, we have $\log r(H) = \log n + O(1)$. This characterization is nearly optimal: for every $\Delta$ there exists an $n$-vertex bipartite graph $H$ of degree at most $\Delta$ and $b(H) \leq \exp(-c\Delta)\,n$, such that $\log r(H) = \log n + \Omega(\Delta)$. We also provide bounds interpolating between these two bandwidth regimes.