Resolution of the Kohayakawa-Kreuter conjecture

A graph G is said to be Ramsey for a tuple of graphs (H_1,…,H_r) if every r-coloring of the edges of G contains a monochromatic copy of H_i in color i, for some i. A fundamental question at the intersection of Ramsey theory and the theory of random graphs is to determine the threshold at which the binomial random graph G_n,p becomes a.a.s. Ramsey for a fixed tuple (H_1,…,H_r), and a famous conjecture of Kohayakawa and Kreuter predicts this threshold. Earlier work of Mousset-Nenadov-Samotij, Bowtell-Hancock-Hyde, and Kuperwasser-Samotij-Wigderson has reduced this probabilistic problem to a deterministic graph decomposition conjecture. In this paper, we resolve this deterministic problem, thus proving the Kohayakawa-Kreuter conjecture. Along the way, we prove a number of novel graph decomposition results which may be of independent interest.