Geometric triangle-free graphs of large chromatic number

Abstract

We present several geometric constructions of triangle-free and large girth families of graphs with rapidly growing chromatic numbers. 1. We construct a triangle-free intersection graph of $n$ boxes in $\mathbb{R}^3$ with independence number $n(\log n)^{-1+o(1)}$, and thus chromatic number $(\log n)^{1-o(1)}$. This is the first improvement over the double logarithmic lower bound of Burling from 1965, and almost matches the best known upper bound $O(\log n)$. Moreover, the bound $o(n)$ on the independence number answers a question of Walczak. 2. We prove that if there exists a unit distance graph in $\mathbb{R}^d$ of chromatic number $r$, then there also exists an induced unit distance graph in $\mathbb{R}^d$ of girth at least $g$ and the same chromatic number. Thus, the problem of determining the maximum chromatic number of unit distance graphs with any prescribed lower bound on the girth reduces entirely to the unrestricted problem, thereby strengthening a long line of results. 3. We construct an intersection graph of $n$ lines in $\mathbb{R}^3$ with girth at least $g$ and chromatic number $Ω_g((\log n)^{1-o(1)})$. This quantitatively improves a construction of Davies. 4. We construct a set of $n$ circles in the plane such that the tangency graph of the circles has girth at least $g$ and chromatic number $Ω_g((\log n)^{1-o(1)})$. This quantitatively improves the result of Davies, Keller, Kleist, Smorodinsky, and Walczak, and provides an alternative solution of Ringel's circle problem. 5. We construct triangle-free ordered graphs on $n$ vertices avoiding a fixed ordered path of length three as an induced subgraph, and having chromatic number $n^{Ω(1/\log \log n)}$. This is motivated by ordered analogues of the Gyárfás-Sumner conjecture.