Given $n$, the number of vertices, what is the number of triangle-free simple graphs on $n$ vertices (or asymptotically)?


A more difficult problem is, given $n$, $m$, what is the number of triangle-free simple graphs on $n$ vertices with $\le m$ edges (or asymptotically)?



counting triangle free graphs
Next terms are 581460254001, 31720840164950 (sent to OEIS). All these numbers were found by exhaustive enumeration. As far as I know, the theoretic enumeration problem is unsolved, even for labelled graphs. 


As for asymptotics, it is an old result of Erdős, Kleitman and Rothschild that almost all triangle-free graphs are bipartite. Someone studied the case of restricted numbers of edges see <http://onlinelibrary.wiley.com/doi/10.1002/(SICI)1097-0118(199602)21:2%3C137::AID-JGT3%3E3.0.CO;2-S/abstract> . 


