Resolution of the Kohayakawa–Kreuter conjecture

A graph 𝐺 is said to be Ramsey for a tuple of graphs(𝐻 1 , … , 𝐻𝑟 ) if every 𝑟-coloring of the edges of 𝐺 con-tains a monochromatic copy of 𝐻𝑖 in color 𝑖, for some 𝑖.A fundamental question at the intersection of Ramseytheory and the theory of random graphs is to deter-mine the threshold at which the binomial randomgraph 𝐺𝑛,𝑝 becomes asymptotically almost surely Ram-sey for a fixed tuple (𝐻 1 , … , 𝐻𝑟 ), and a famous conjectureof Kohayakawa and Kreuter predicts this threshold.Earlier work of Mousset–Nenadov–Samotij, Bowtell–Hancock–Hyde, and Kuperwasser–Samotij–Wigdersonhas reduced this probabilistic problem to a determinis-tic graph decomposition conjecture. In this paper, weresolve this deterministic problem, thus proving theKohayakawa–Kreuter conjecture. Along the way, weprove a number of novel graph decomposition resultsthat may be of independent interest.