Asymmetric Ramsey properties of randomly perturbed graphs

Abstract

In this note, we investigate for various pairs of graphs $(H,G)$ the question of how many random edges must be added to a dense graph to guarantee that any red-blue coloring of the edges contains a red copy of $H$ or a blue copy of $G$. We determine this perturbed Ramsey threshold for many new pairs of graphs and various ranges of densities, obtaining several generalizations of results obtained by Das and Treglown. In particular, we resolve the remaining cases toward determining the perturbed Ramsey threshold for pairs $(K_t,K_s)$ where $t\geq s\geq 5$.