TITLE: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices
QUESTION [18 upvotes]: Among the collections of the open problems of Paul Erdős on the website of
Professor Fan Chung, there is one called "number of triangle-free graphs".
http://www.math.ucsd.edu/~erdosproblems/erdos/newproblems/NumberOfTriangleFreeGraphs.html
Open Problem: Determine or estimate the number of maximal triangle-free graphs
on $n$ vertices.
Is any one working on this problem? Any known related results? 
Now I am considering about using a "connection game" method to solve this problem: Given a set of $n$ players, each one chooses to connect to other nodes, if any two neighbors of it connect a new edge(which means there would form a triangle), it has to choose delete either edges with those two neighbors. Then the question is how many different topology connections does it have? 
Any comments on this method?

REPLY [5 votes]: This question was recently solved by Balogh & Petrickova. Douglas Zare's bound is tight (apart from the $o(1)$ term). See http://arxiv.org/abs/1409.8123