  In this article we consider a problem related to two famous combinatorial
topics. One of them concerns the chromatic number of the space. The other deals
with graphs having big girth (the length of the shortest cycle) and large
chromatic number. Namely, we prove that for any $l\in \mathbb{N}$ there exists
a sequence of distance graphs in $\mathbb{R}^n$ with girth at least $l$ and the
chromatic number equal to $(c+\bar{o}(1))^n$ with $c>1$.
