I recall that I once read a short paper on the following subject: Fix a graph $H$ with fractional coloring number $c$, now let $G$ be a graph with $n$ vertices and $e$ edges, with $e$,$n$ large, we want to bound from below and above the number of homomorphisms from $H$ to $G$ in terms of $n,e,c$.
I now cannot seem to find the paper, can someone help me?
I don't remember the authors, but they mentioned a thanks to Gil Kalai for suggesting the connection with fractional coloring number is a detail I remember for some reason.