Improved Ramsey-type results for comparability graphs

TL;DR

This paper improves bounds on the size of homogeneous sets in graphs formed by unions of comparability graphs, showing near-optimal results for the case of two such graphs using probabilistic constructions.

Contribution

It introduces a probabilistic method to construct graphs from unions of comparability graphs with smaller homogeneous sets, improving known bounds for the case of two graphs.

Findings

Constructed two comparability graphs with union lacking large cliques or independent sets.

Extended the approach to unions of r comparability graphs, limiting bipartite subgraphs.

Improved bounds on homogeneous set sizes in union graphs, refining previous results.

Abstract

Several discrete geometry problems are equivalent to estimating the size of the largest homogeneous sets in graphs that happen to be the union of few comparability graphs. An important observation for such results is that if $G$ is an $n$-vertex graph that is the union of $r$ comparability (or more generally, perfect) graphs, then either $G$ or its complement contains a clique of size $n^{1/(r+1)}$. This bound is known to be tight for $r=1$. The question whether it is optimal for $r\ge 2$ was studied by Dumitrescu and T\'oth. We prove that it is essentially best possible for $r=2$, as well: we introduce a probabilistic construction of two comparability graphs on $n$ vertices, whose union contains no clique or independent set of size $n^{1/3+o(1)}$. Using similar ideas, we can also construct a graph $G$ that is the union of $r$ comparability graphs, and neither $G$, nor its complement contains a complete bipartite graph with parts of size $\frac{cn}{(\log n)^r}$. With this, we improve a result of Fox and Pach.