TITLE: Explicit graphs with large chromatic number and girth
QUESTION [7 upvotes]: It is well known that there exist graphs with large chromatic number and girth. More precisely, for any $k$ and $l$, there exists a graph $G$ such that $\chi(G) > k$ (where $\chi$ denotes the chromatic number) and $G$ does not contain $C_m$ (the cycle of length $m$) for any $3 \leq m \leq l$. 
The standard proof of this theorem (due to Erdos) is one of the earliest and  arguably most important applications of the probabilistic method. One basically shows that a random graph $G_{n,p}$ (for large $n$ and carefully chosen $p$) can be modified by removing a small proportion of the vertices to obtain a graph with no short cycles and no large independent sets - hence a graph with large girth and large chromatic number.
Another thing which is well known is that triangle-free graphs with large chromatic number exist. These graphs can be constructed very explicitly using a construction of Mycielski.
The following question is thus very natural, and I suspect the answer is well known to many (but unfortunately not me):

Does there exist an explicitly defined family $G_{k,l}$ with $\chi(G_{k,l}) \geq k$ and $\mathrm{girth}(G_{k,l}) \geq l$ ?

I'm tagging this as reference request, since I'm sure the question has received much attention from experts. Ideally, I would hope for either a paper which provides such a construction, or a source suggesting that no such construction is presently known.

REPLY [3 votes]: According to the graph theoretical monograph of Diestel, "Lovász (1968) was also the first to construct graphs of arbitrarily large girth and chromatic number". I've checked on mathscinet and it seems that the reference is to this article.