Orientation Ramsey thresholds for cycles and cliques

TL;DR

This paper determines the probability thresholds at which random graphs almost surely contain certain oriented subgraphs, specifically acyclic orientations of complete graphs and cycles, in every edge orientation.

Contribution

It establishes the threshold functions for the property that all orientations of a random graph contain a given oriented cycle or clique.

Findings

Thresholds for oriented cycles in random graphs are identified.

Thresholds for oriented cliques in random graphs are established.

Results apply to all orientations of the specified subgraphs.

Abstract

If $G$ is a graph and $\vec H$ is an oriented graph, we write $G\to \vec H$ to say that every orientation of the edges of $G$ contains $\vec H$ as a subdigraph. We consider the case in which $G=G(n,p)$, the binomial random graph. We determine the threshold $p_{\vec H}=p_{\vec H}(n)$ for the property $G(n,p)\to \vec H$ for the cases in which $\vec H$ is an acyclic orientation of a complete graph or of a cycle.