The number of the maximal triangle-free graphs
  Paul Erd\H{o}s suggested the following problem: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. Here we show that the number of maximal triangle-free graphs is at most $2^{n^2/8+o(n^2)}$, which matches the previously known lower bound. Our proof uses among others the Ruzsa-Szemer\'{e}di triangle removal lemma, and recent results on characterizing of the structure of independent sets in hypergraphs. 
J\'ozsef Balogh and \v{S}\'arka Pet\v{r}\'i\v{c}kov\'a
math.CO