Ramsey properties of semilinear graphs

TL;DR

This paper establishes that semilinear graphs of constant complexity exhibit very controlled Ramsey properties, with bounds on clique and independent set sizes, and provides new bounds on their chromatic number and symmetric Ramsey properties.

Contribution

It proves that semilinear graphs of constant complexity have tight bounds on clique and independent set sizes, and introduces new results on their chromatic number and symmetric Ramsey properties.

Findings

Semilinear graphs of constant complexity have at most O_{s,t}(n)(log n)^{O_t(1)} vertices without large cliques or independent sets.

Proper coloring of such graphs can be achieved with polylogarithmic number of colors.

The paper extends known bounds on intersection graphs of geometric objects to a broader class of semilinear graphs.

Abstract

A graph $G$ is semilinear of complexity $t$ if the vertices of $G$ are elements of $\mathbb{R}^{d}$ for some $d\in\mathbb{Z}^{+}$, and the edges of $G$ are defined by the sign patterns of $t$ linear functions $f_1,\dots,f_t:\mathbb{R}^{d}\times \mathbb{R}^{d}\rightarrow\mathbb{R}$. We show that semilinear graphs of constant complexity have very tame Ramsey properties. More precisely, we prove that if $G$ is a semilinear graph of complexity $t$ which contains no clique of size $s$ and no independent set of size $n$, then $G$ has at most $O_{s,t}(n)\cdot(\log n)^{O_t(1)}$ vertices. We also show that the logarithmic term cannot be omitted. In particular, this implies that if $G$ is a semilinear graph of constant complexity on $n$ vertices, and $G$ contains no clique of size $s$, then $G$ can be properly colored with $\mbox{polylog}(n)$ colors. In the past 60 years, this coloring question was extensively studied for several special instances of semilinear graphs, e.g. shift graphs, intersection and disjointness graphs of certain geometric objects, and overlap graphs. Our main result provides a general upper bound on the chromatic number of all such, seemingly unrelated, graphs. Furthermore, we consider the symmetric Ramsey problem for semilinear graphs as well. It is known that if there exists an intersection graph of $N$ boxes in $\mathbb{R}^{d}$ (such graphs are semilinear of complexity $2d$) that contains no clique or independent set of size $n$, then $N=O_d(n^2(\log n)^{d-1})$. That is, the exponent of $n$ does not grow with the dimension. We prove a result about the symmetric Ramsey properties of semilinear graphs, which puts this phenomenon in a more general context.