Towards the Kohayakawa--Kreuter conjecture on asymmetric Ramsey properties

TL;DR

This paper proves an upper bound on the threshold function for asymmetric Ramsey properties in random graphs, confirming a key conjecture by Kohayakawa and Kreuter.

Contribution

It establishes the 1-statement of the Kohayakawa--Kreuter conjecture regarding asymmetric Ramsey thresholds.

Findings

Proved an upper bound on the threshold function for asymmetric Ramsey properties.

Confirmed the 1-statement of the Kohayakawa--Kreuter conjecture.

Advances understanding of phase transitions in random graphs for multiple fixed subgraphs.

Abstract

For fixed graphs $F_1,\dotsc,F_r$, we prove an upper bound on the threshold function for the property that $G(n,p) \to (F_1,\dotsc,F_r)$. This establishes the $1$-statement of a conjecture of Kohayakawa and Kreuter.