Proof of the Kohayakawa--Kreuter conjecture for the majority of cases

Candida Bowtell; Robert Hancock; Joseph Hyde

Paper

Proof of the Kohayakawa--Kreuter conjecture for the majority of cases

Abstract

For graphs

, write

to denote the property that whenever we

-colour the edges of

, there is a monochromatic copy of

in colour

for some

. Mousset, Nenadov and Samotij proved an upper bound on the threshold function for the property that

, thereby resolving the

-statement of the Kohayakawa--Kreuter conjecture. We reduce the

-statement of the Kohayakawa--Kreuter conjecture to a natural deterministic colouring problem and resolve this problem for almost all cases, which in particular includes (but is not limited to) when

is strictly

-balanced and either has density greater than

or is not bipartite. In addition, we extend our reduction to hypergraphs, proving the colouring problem in almost all cases there as well.