  In 1975 Erd\H{o}s initiated the study of the following very natural question.
What can be said about the chromatic number of unit distance graphs in
$\mathbb{R}^2$ that have large girth? Over the years this question and its
natural extension to $\mathbb{R}^d$ attracted considerable attention with the
high-dimensional variant reiterated recently by Alon and Kupavskii.
  We prove that there exist unit distance graphs in $\mathbb{R}^d$ 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; Sagdeev; and Sagdeev
and 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.
