The threshold for the constrained Ramsey property

TL;DR

This paper investigates the threshold probability for the constrained Ramsey property in random graphs, specifically when the second graph is a forest, providing explicit and implicit threshold results.

Contribution

It determines the threshold for the constrained Ramsey property in random graphs when the second graph is a forest, extending understanding of Ramsey properties in probabilistic settings.

Findings

Threshold explicitly determined when it is (n^{-1})

Implicit threshold results for other cases

Characterization of when the constrained Ramsey number exists

Abstract

Given graphs $G$, $H_1$, and $H_2$, let $G\xrightarrow{\text{mr}}(H_1,H_2)$ denote the property that in every edge colouring of $G$ there is a monochromatic copy of $H_1$ or a rainbow copy of $H_2$. The constrained Ramsey number, defined as the minimum $n$ such that $K_n\xrightarrow{\text{mr}}(H_1,H_2)$, exists if and only if $H_1$ is a star or $H_2$ is a forest. We determine the threshold for the property $G(n,p)\xrightarrow{\text{mr}}(H_1,H_2)$ when $H_2$ is a forest, explicitly when the threshold is $\Omega(n^{-1})$ and implicitly otherwise.