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?