Ramsey-type problems in orientations of graphs

TL;DR

This paper investigates Ramsey-type problems for orientations of graphs, establishing bounds on the minimum graph size guaranteeing an oriented subgraph, analyzing thresholds in random graphs, and exploring isometric variants using advanced combinatorial techniques.

Contribution

The paper provides new bounds on oriented Ramsey numbers, studies threshold functions in random graphs, and introduces isometric Ramsey number bounds for acyclic orientations of cycles.

Findings

Bound $oldsymbol{oldsymbol{ ext{R}(oldsymbol{oldsymbol{ ext{H}}})}}$ in terms of classical Ramsey numbers.

Determines threshold functions for the property $oldsymbol{oldsymbol{G(n,p) o oldsymbol{oldsymbol{ ext{H}}}}}$ in random graphs.

Establishes upper bounds for isometric Ramsey numbers of acyclic orientations of cycles.

Abstract

Given an acyclic oriented graph $\vec{H}$ and a graph $G$, we write $G \to \vec{H}$ if every orientation of $G$ has an oriented copy of $\vec{H}$. We define $\vec{R}(\vec{H})$ as the smallest number $n$ such that there exists a graph $G$ satisfying $G \to \vec{H}$. Denoting by $R(H)$ the classical Ramsey number of a graph $H$, we show that $\vec{R}(\vec{H}) \leq 2R(H)^{c \log^2 h}$ for every acyclic oriented graph $\vec{H}$ with $h$ vertices, where $H$ is its underlying undirected graph. We also study the threshold function for the event $\{G(n,p) \to \vec{H}\}$ in the binomial random graph $G(n,p)$. Finally, we consider the isometric case, in which we require that, for every two vertices $x, y \in V(\vec{H})$ and their respective copies $x', y'$ in $\vec{G}$, the distance between $x$ and $y$ is equal to the distance between $x'$ and $y'$. We prove an upper bound for the isometric Ramsey number of an acyclic orientation of the cycle, applying the hypergraph container lemma in random graphs.