Ramsey properties 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.