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?

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?