The List-Ramsey threshold for families of graphs

Given a family of graphs F and an integer r, we say that a graph is r-Ramsey for F if any r-colouring of its edges admits a monochromatic copy of a graph from F. The threshold for the classic Ramsey property, where 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 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.