Towards the 0-statement of the Kohayakawa-Kreuter conjecture

TL;DR

This paper advances understanding of asymmetric Ramsey properties in random graphs by reducing a key conjecture's 0-statement to a deterministic subproblem and solving it for most regular graph pairs.

Contribution

It reduces the 0-statement of the Kohayakawa-Kreuter conjecture to a deterministic subproblem and solves it for almost all pairs of regular graphs, advancing the conjecture's resolution.

Findings

Reduces the 0-statement to a deterministic subproblem.

Solves the subproblem for almost all pairs of regular graphs.

Progress towards resolving the asymmetric Ramsey threshold conjecture.

Abstract

In this paper, we study asymmetric Ramsey properties of the random graph $G_{n,p}$. Let $r \in \mathbb{N}$ and $H_1, \ldots, H_r$ be graphs. We write $G_{n,p} \to (H_1, \ldots, H_r)$ to denote the property that whenever we colour the edges of $G_{n,p}$ with colours from the set $[r] := \{1, \ldots, r\}$ there exists $i \in [r]$ and a copy of $H_i$ in $G_{n,p}$ monochromatic in colour $i$. There has been much interest in determining the asymptotic threshold function for this property. R\"{o}dl and Ruci\'{n}ski determined the threshold function for the general symmetric case; that is, when $H_1 = \cdots = H_r$. A conjecture of Kohayakawa and Kreuter, if true, would fully resolve the asymmetric problem. Recently, the 1-statement of this conjecture was confirmed by Mousset, Nenadov and Samotij. Building on work of Marciniszyn, Skokan, Sp\"{o}hel and Steger, 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.