Number of triangle-free graphs

Erdös, Kleitman and Rothschild \( ^{[2]}\) proved that almost all triangle-free graphs on \( n\) vertices are bipartite. A triangle-free graph \( G\) is said to be maximal if adding any edge to \( G\) will result in a triangle. A problem on maximal triangle-free graphs is the following:

Problem [1]

Determine or estimate the number of maximal triangle-free graphs on \(n\) vertices.

Bibliography
1 M. Simonovits, Paul Erdös' influence on extremal graph theory, The Mathematics of Paul Erdös, II (R. L. Graham and J. Nešetril, eds.), 148-192, Springer-Verlag, Berlin, 1996.

2 P. Erdös, D. J. Kleitman and B. L. Rothschild, Asymptotic enumeration of \( K_n\)-free graphs (Italian summary), Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973), Tomo II, Atti dei Convegni Lincei, No. 17, 19-27, Accad. Naz. Lincei, Rome, 1976.