Geometric Graphs with Exponential Chromatic Number and Arbitrary Girth
Matija Bucić, James Davies · American Mathematical Monthly · 2025
In 1975 Paul Erdős initiated the study of the following very natural question. What can be said about the chromatic number of unit distance graphs in R2 that have large girth? Over the years this question and its natural extension to Rd attracted considerable attention with the high-dimensional variant reiterated recently by Noga Alon and Andrey Kupavskii.We prove that there exist unit distance graphs in Rd with chromatic number at least (1.074+o(1))d that have arbitrarily large girth. This improves upon a series of results due to Kupavskii; Arsenii Sagdeev; and Sagdeev and Andrei Raigorodskii and gives the first bound in which the base of the exponent does not tend to one with the girth. In addition, our construction can be made explicit which allows us to answer in a strong form a question of Kupavskii.Our arguments show graphs of large chromatic number and high girth exist in a number of other geometric settings, including diameter graphs and orthogonality graphs.