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, where $\mathcal{F}$ consists of one graph, in the binomial random graph was located in the celebrated work of Rödl and Ruciński. In this paper, we offer a twofold generalisation to the Rödl–Ruciński 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 Rödl–Ruciński theorem.