Graph Classes With Linear Ramsey Numbers

The Ramsey number R-X (p, q) for a class of graphs X is the minimum n such that every graph in X with at least n vertices has either a clique of size p or an independent set of size q. We say that Ramsey numbers are linear in X if there is a constant k such that R-X (p, q) <= k(p + q) for all p, q. In the present paper we conjecture that if X is a hereditary class defined by finitely many forbidden induced subgraphs, then Ramsey numbers are linear in X if and only if X excludes a forest, a disjoint union of cliques and their complements. We prove the "only if'' part of this conjecture and verify the "if'' part for a variety of classes. We also apply the notion of linearity to bipartite Ramsey numbers and reveal a number of similarities and differences between the bipartite and non-bipartite case. (C) 2021 Elsevier B.V. All rights reserved.