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Paul Erdős suggested the following problem: Determine or estimate the number of maximal triangle‐free graphs on   n   vertices. Here we show that the number of maximal triangle‐free graphs is at most    2    n  2   /  8  +  o   (   n  2   ), which matches the previously known lower bound.  Our proof uses among others the Ruzsa–Szemerédi triangle‐removal lemma, and recent results on characterizing of the structure of independent sets in hypergraphs.

The number of the maximal triangle‐free graphs