Counterexamples to Borsuk's Conjecture with Large Girth

Borsuk's celebrated conjecture, which has been disproved, can be stated as follows: in $\mathbb R^n$, there exist no diameter graphs with chromatic number larger than $n+1$. In this paper, we prove the existence of counterexamples to Borsuk's conjecture which, in addition, have large girth. This study is in the spirit of O'Donnell and Kupavskii, who studied the existence of distance graphs with large girth. We consider both cases of strict and nonstrict diameter graphs. We also prove the existence of counterexamples with large girth to a statement of Lovász concerning distance graphs on the sphere.