On the Kohayakawa-Kreuter conjecture

TL;DR

This paper proves the Kohayakawa-Kreuter conjecture for most graph tuples, establishing the threshold for when a random graph almost surely becomes Ramsey for those graphs, and links the conjecture to a deterministic graph partitioning problem.

Contribution

It resolves the conjecture for nearly all graph tuples and reduces its proof to verifying a specific deterministic statement, broadening the conjecture's applicability.

Findings

Proves the conjecture for almost all graph tuples.

Reduces the conjecture to a deterministic graph partitioning problem.

Introduces a new graph-partitioning conjecture of independent interest.

Abstract

Let us say that a graph $G$ is Ramsey for a tuple $(H_1,\dots,H_r)$ of graphs if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$, for some $i \in [r]$. A famous conjecture of Kohayakawa and Kreuter, extending seminal work of R\"odl and Ruci\'nski, predicts the threshold at which the binomial random graph $G_{n,p}$ becomes Ramsey for $(H_1,\dots,H_r)$ asymptotically almost surely. In this paper, we resolve the Kohayakawa-Kreuter conjecture for almost all tuples of graphs. Moreover, we reduce its validity to the truth of a certain deterministic statement, which is a clear necessary condition for the conjecture to hold. All of our results actually hold in greater generality, when one replaces the graphs $H_1,\dots,H_r$ by finite families $\mathcal{H}_1,\dots,\mathcal{H}_r$. Additionally, we pose a natural (deterministic) graph-partitioning conjecture, which we believe to be of independent interest, and whose resolution would imply the Kohayakawa-Kreuter conjecture.