Existence of graph with large chromatic number and girth

Question

Show that for each constant $c$, for sufficiently large $n$ (with respect to $c$) there is a graph $G=(V,E)$ with $n$ vertices and bounded (constant which depends on $c$) degree $d$ such that $G$ has no coloring of size $c$ AND $G$ has no cycles of length smaller than $O(\log(n))$.
My idea is to randomly choose appropriate graph drawn from $G(n,p)$ and then remove the "exceeding" edges in order to get bounded degree - but if I do so the chromatic number may change.
Any ideas?

Answer 1

It is known that there are triangle-free graphs with arbitrary large chromatic number, so called Mycielski graphs. This idea was further generalized to build graphs with higher girth and arbitrary high chromatic number.