A Sparse Transference Principle for a Non-Monotone Ramsey Property

Abstract

We prove a sparse transference theorem for induced Ramsey graphs. The theorem transfers the weighted random-host proof of Aragão, Campos, Dahia, Filipe, and Marciano to the sparse random setting. It follows that, for every fixed graph $H$ with no isolated vertices and at least two edges, and every $\hat{I}\cdot >0$, there is $C>0$ such that, whenever $N\ge r^{Cr}$ and $N^{-1/m_2(H)+\hat{I}\cdot }\le p\le \frac{1}{2}$, with high probability every $r$-colouring of the edges of $G(N,p)$ contains a monochromatic induced copy of $H$. Here, $m_2(H)$ denotes the usual maximum 2-density of $H$.