On induced Ramsey numbers for multiple copies of graphs

TL;DR

This paper investigates the relationship between induced Ramsey numbers for pairs of graphs and their multiple copies, revealing cases where the known inequality is strict and others where equality holds, thus advancing understanding of graph coloring properties.

Contribution

The paper demonstrates that the inequality IR(sG, tH) ≤ (s+t-1)IR(G,H) can be strict for certain graph classes, providing new insights into induced Ramsey numbers for multiple graph copies.

Findings

Identifies infinite graph classes where the inequality is strict.

Shows IR(sG, tH) can be arbitrarily smaller than (s+t-1)IR(G,H).

Provides examples where the inequality holds with equality.

Abstract

We say that a graph F strongly arrows a pair of graphs (G,H) if any colouring of its edges with red and blue leads to either a red G or a blue H appearing as induced subgraphs of F. The induced Ramsey number, IR(G,H) is defined as the smallest order of a graph that strongly arrows (G,H). We consider the connection between the induced Ramsey number for a pair of two connected graphs IR(G,H) and the induced Ramsey number for multiple copies of these graphs IR(sG,tH), where xG denotes the pairwise vertex-disjoint union of x copies of G. It is easy to see that if F strongly arrow (G,H), then (s+t-1)F strongly arrows (sG, tH). This implies that IR(sG, tH) is at most (s+t-1)IR(G,H). For all known results on induced Ramsey numbers for multiple copies, the inequality above holds as equality. We show that there are infinite classes of graphs for which the inequality above is strict and moreover, IR(sG, tH) could be arbitrarily smaller than (s+t-1)IR(G,H). On the other hand, we provide further examples of classes of graphs for which the inequality above holds as equality.