Size-Ramsey numbers of structurally sparse graphs

TL;DR

This paper establishes new bounds on size-Ramsey numbers for graphs with bounded treewidth, extending previous results from constant degree and treewidth to more general classes like planar graphs.

Contribution

It generalizes existing bounds on size-Ramsey numbers to graphs with growing treewidth, including minor-closed classes such as planar graphs.

Findings

Bound of O(n^{3/2 - 1/2}) for minor-closed classes

Addresses open question for planar graphs

Combines structural graph theory with Ramsey techniques

Abstract

Size-Ramsey numbers are a central notion in combinatorics and have been widely studied since their introduction by Erd\H{o}s, Faudree, Rousseau and Schelp in 1978. Research has mainly focused on the size-Ramsey numbers of $n$-vertex graphs with constant maximum degree $\Delta$. For example, graphs which also have constant treewidth are known to have linear size-Ramsey numbers. On the other extreme, the canonical examples of graphs of unbounded treewidth are the grid graphs, for which the best known bound has only very recently been improved from $O(n^{3/2})$ to $O(n^{5/4})$ by Conlon, Nenadov and Truji\'c. In this paper, we prove a common generalization of these results by establishing new bounds on the size-Ramsey numbers in terms of treewidth (which may grow as a function of $n$). As a special case, this yields a bound of $\tilde{O}(n^{3/2 - 1/2\Delta})$ for proper minor-closed classes of graphs. In particular, this bound applies to planar graphs, addressing a question of Kamcev, Liebenau, Wood and Yepremyan. Our proof combines methods from structural graph theory and classic Ramsey-theoretic embedding techniques, taking advantage of the product structure exhibited by graphs with bounded treewidth.