On the Kohayakawa–Kreuter conjecture

Let us say that a graph G is Ramsey for a tuple (H1, ... , Hr) of graphs if every r-colouring of the edges of G contains a monochromatic copy of Hi in colour i, for some i ∈ [[r]]. A famous conjecture of Kohayakawa and Kreuter, extending seminal work of Rödl and Rucinski, predicts the threshold at which the binomial random graph ´ Gn,p becomes Ramsey for (H1, ... , Hr) asymptotically almost surely. In this paper, we resolve the Kohayakawa–Kreuter conjecture for almost all tuples of graphs. Moreover, we reduce its validity to the truth of a certain deterministic statement, which is a clear necessary condition for the conjecture to hold. All of our results actually hold in greater generality, when one replaces the graphs H1, ... , Hr by finite families H1, ... , Hr. Additionally, we pose a natural (deterministic) graph-partitioning conjecture, which we believe to be of independent interest, and whose resolution would imply the Kohayakawa–Kreuter conjecture.