Vertex Ramsey properties of randomly perturbed graphs

TL;DR

This paper investigates how adding a small number of random edges to a dense graph guarantees the vertex Ramsey property for pairs involving at least one clique, extending previous threshold results to perturbed graphs.

Contribution

It determines the minimum number of random edges needed to ensure vertex Ramsey properties in perturbed graphs for pairs involving cliques, a novel extension of existing threshold results.

Findings

Identifies the threshold of random edges for vertex Ramsey properties in perturbed graphs.

Extends classical results from purely random graphs to perturbed dense graphs.

Provides probabilistic bounds ensuring the Ramsey property with high probability.

Abstract

Given graphs $F,H$ and $G$, we say that $G$ is $(F,H)_v$-Ramsey if every red/blue vertex colouring of $G$ contains a red copy of $F$ or a blue copy of $H$. Results of {\L}uczak, Ruci\'nski and Voigt and, subsequently, Kreuter determine the threshold for the property that the random graph $G(n,p)$ is $(F,H)_v$-Ramsey. In this paper we consider the sister problem in the setting of randomly perturbed graphs. In particular, we determine how many random edges one needs to add to a dense graph to ensure that with high probability the resulting graph is $(F,H)_v$-Ramsey for all pairs $(F,H)$ that involve at least one clique.