Article contents
Improved Ramsey-type results for comparability graphs
Published online by Cambridge University Press: 13 August 2020
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 Tóth. 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 contain a complete bipartite graph with parts of size $cn/{(log n)^r}$ . With this, we improve a result of Fox and Pach.
MSC classification
- Type
- Paper
- Information
- Copyright
- © The Author(s), 2020. Published by Cambridge University Press
Footnotes
Research supported in part by SNSF grants 200020-162884 and 200021-175977.
References
- 1
- Cited by