Upper bounds on probability thresholds for asymmetric Ramsey properties

TL;DR

This paper establishes upper bounds on the probability thresholds for asymmetric Ramsey properties in random graphs, confirming parts of a conjecture and providing an alternative proof technique to the deletion method.

Contribution

It introduces a new upper bound for the threshold in asymmetric Ramsey properties and offers an alternative to the deletion method used in previous proofs.

Findings

Established the order of magnitude of thresholds for complete graphs.

Partially confirmed a conjecture by the authors and Kreuter.

Provided an alternative proof technique to the deletion method.

Abstract

Given two graphs $G$ and $H$, we investigate for which functions $p=p(n)$ the random graph $G_{n,p}$ (the binomial random graph on $n$ vertices with edge probability $p$) satisfies with probability $1-o(1)$ that every red-blue-coloring of its edges contains a red copy of $G$ or a blue copy of $H$. We prove a general upper bound on the threshold for this property under the assumption that the denser of the two graphs satisfies a certain balancedness condition. Our result partially confirms a conjecture by the first author and Kreuter, and together with earlier lower bound results establishes the exact order of magnitude of the threshold for the case in which $G$ and $H$ are complete graphs of arbitrary size. In our proof we present an alternative to the so-called deletion method, which was introduced by R\"odl and Ruci\'{n}ski in their study of symmetric Ramsey properties of random graphs (i.e. the case $G=H$), and has been used in many proofs of similar results since then.