Ramsey properties for tilings in random graphs

TL;DR

This paper extends classical Ramsey theory results to random graphs, establishing the threshold probability for monochromatic tilings in edge-coloured random graphs, generalizing prior work for most graphs.

Contribution

It proves the threshold for monochromatic tilings in random graphs matches known bounds for most graphs, extending previous results to the case of two colours.

Findings

Established the threshold $n^{-1/ ext{max}\{m_2(H),1"}}$ for monochromatic tilings in random graphs.

Generalized Burr, Erdős, and Spencer's classical result to random graph settings.

Extended R"{o}dl and Ruciński's threshold results to the case of two colours.

Abstract

Let $mH$ be the graph formed by $m$ vertex-disjoint copies of a graph $H$. Let $G \to (H)_r$ denote that, in any $r$-colouring of the edges of $G$, there exists a monochromatic copy of $H$. In 1975, Burr, Erd\H{o}s, and Spencer showed that if $H$ is a graph on $k$ vertices whose independence number is $\alpha$, then $K_n \to (mH)_2$, where $m\sim n/(2k-\alpha)$, and that the $1/(2k-\alpha)$ factor is best possible. In the 1990s, R\"{o}dl and Ruci\'{n}ski proved that, for all but a few graphs~$H$, the threshold for the property $\mathbb{G}(n,p) \to (H)_r$ is $n^{-1/m_2(H)}$. In this paper, generalizing the result of Burr, Erd\H{o}s, and Spencer, we prove that $n^{-1/\max\{m_2(H),1\}}$ is the threshold for the property $\mathbb{G}(n,p) \to (mH)_2$, where $m\sim n/(2k-\alpha)$. This threshold matches the one found by R\"{o}dl and Ruci\'nski for most graphs $H$, extending their result in the case $r=2$.