The number of the maximal triangle-free graphs

TL;DR

This paper establishes an upper bound on the number of maximal triangle-free graphs on n vertices, matching the known lower bound, using advanced combinatorial tools like the triangle removal lemma.

Contribution

It provides a tight asymptotic estimate for the number of maximal triangle-free graphs, combining combinatorial and hypergraph structural results.

Findings

Number of maximal triangle-free graphs is at most 2^{n^2/8+o(n^2)}

The upper bound matches the known lower bound, confirming the asymptotic count

Utilizes the Ruzsa-Szemerédi triangle removal lemma and hypergraph structure results.

Abstract

Paul Erd\H{o}s 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-Szemer\'{e}di triangle removal lemma, and recent results on characterizing of the structure of independent sets in hypergraphs.