The list-Ramsey threshold for families of graphs

Abstract

Given a family of graphs $\mathcal{F}$ and an integer $r$, we say that a graph is $r$-Ramsey for $\mathcal{F}$ if any $r$-colouring of its edges admits a monochromatic copy of a graph from $\mathcal{F}$. The threshold for the classic Ramsey property in the binomial random graph, where $\mathcal{F}$ consists of one graph, was located in the celebrated work of R\"odl and Ruci\'nski. In this paper, we offer a twofold generalisation to the R\"odl--Ruci\'nski theorem. First, we show that the list-colouring version of the property has the same threshold. Second, we extend this result to finite families $\mathcal{F}$, where the threshold statements might also diverge. This also confirms further special cases of the Kohayakawa--Kreuter conjecture. Along the way, we supply a short(-ish), self-contained proof of the $0$-statement of the R\"odl--Ruci\'nski theorem.