Towards the 0-statement of the Kohayakawa-Kreuter conjecture

Abstract

In this paper, we study asymmetric Ramsey properties of the random graph Gn,p. Let r ∈ N and H1, . . . , Hr be graphs.We write Gn,p→(H1, . . . , Hr) to denote the property that whenever we colour the edges of Gn,p with colours from the set [r] := {1, . . . , r} there exists i ∈ [r] and a copy of Hi in Gn,p monochromatic in colour i. There has been much interest in determining the asymptotic threshold function for this property. In several papers, Rödl and Ruci´nski determined a threshold function for the general symmetric case; that is, when H1 = · · · = Hr . A conjecture of Kohayakawa and Kreuter from 1997, if true, would fully resolve the asymmetric problem. Recently, the 1-statement of this conjecture was confirmed byMousset, Nenadov and Samotij. Building on work of Marciniszyn, Skokan, Spöhel and Steger from 2009, we reduce the 0-statement of Kohayakawa and Kreuter’s conjecture to a certain deterministic subproblem. To demonstrate the potential of this approach, we show this subproblem can be resolved for almost all pairs of regular graphs. This therefore resolves the 0-statement for all such pairs of graphs.