Graph classes with linear Ramsey numbers

TL;DR

This paper investigates conditions under which graph classes have linear Ramsey numbers, proving a key part of a conjecture relating linearity to forbidden subgraphs and exploring bipartite cases.

Contribution

It proves the 'only if' part of a conjecture linking linear Ramsey numbers to forbidden subgraphs and verifies the 'if' part for various classes, advancing understanding of graph Ramsey theory.

Findings

Proved the 'only if' part of the conjecture.

Verified the 'if' part for several graph classes.

Identified similarities and differences in bipartite Ramsey numbers.

Abstract

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 $R_{X}(p,q) \leq k(p+q)$ 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.