On the Kohayakawa-Kreuter conjecture

Let us say that a graph is Ramsey for a tuple of graphs if every r-colouring of the edges of G contains a monochromatic copy of in colour i, for some. A famous conjecture of Kohayakawa and Kreuter, extending seminal work of Rödl and Ruciński, predicts the threshold at which the binomial random graph becomes Ramsey for 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 by finite families. 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.