Directed graphs with lower orientation Ramsey thresholds

TL;DR

This paper studies the threshold probability for the Ramsey property in random directed graphs, revealing that the known upper bounds are not always tight and providing specific examples where the threshold differs.

Contribution

It introduces new examples of directed graphs where the classical upper bound for the Ramsey threshold is not tight, especially involving rooted products and certain sparse graphs.

Findings

The upper bound $p_{\vec H}\leq Cn^{-1/m_2(\vec H)}$ is not always the threshold.

Examples include rooted products of orientations of sparse graphs.

Thresholds vary depending on graph structures such as forests and cycles.

Abstract

We investigate the threshold $p_{\vec H}=p_{\vec H}(n)$ for the Ramsey-type property $G(n,p)\to \vec H$, where $G(n,p)$ is the binomial random graph and $G\to\vec H$ indicates that every orientation of the graph $G$ contains the oriented graph $\vec H$ as a subdigraph. Similarly to the classical Ramsey setting, the upper bound $p_{\vec H}\leq Cn^{-1/m_2(\vec H)}$ is known to hold for some constant $C=C(\vec H)$, where $m_2(\vec H)$ denotes the maximum $2$-density of the underlying graph $H$ of $\vec H$. While this upper bound is indeed the threshold for some $\vec H$, this is not always the case. We obtain examples arising from rooted products of orientations of sparse graphs (such as forests, cycles and, more generally, subcubic $\{K_3,K_{3,3}\}$-free graphs) and arbitrarily rooted transitive triangles.