Resolution of the Kohayakawa-Kreuter conjecture

Abstract

A graph $G$ is said to be Ramsey for a tuple of graphs $(H_1,\dots,H_r)$ 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 $(H_1,\dots,H_r)$, 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.