Structure and stability of triangle-free set systems

We define the notion of stability for a monotone property of set systems. This phenomenon encompasses some classical results in combinatorics, foremost among them the Erdos-Simonovits stability theorem. A triangle is a family of three sets A, B, C such that An B, B ∩ C, C ∩ A are each nonempty, and A ∩ B n C = 0. We prove the following new theorem about the stability of triangle-free set systems. Fix r > 3. For every δ > 0, there exist e > 0 and no = n 0 (e, r) such that the following holds for all n > no: if |X| = n and G is a triangle-free family of r-sets of X containing at least (1 - e)( n-1 r-1 ) members, then there exists an (n - 1)-set S C X which contains fewer than δ( n-1 r-1 ) members of G. This is one of the first stability theorems for a nontrivial problem in extremal set theory. Indeed, the corresponding extremal result, that for n > 3r/2 > 4 every triangle-free family G of r-sets of X has size at most ( n-1 r-1 ), was a longstanding conjecture of Erdos (open since 1971) that was only recently settled by Mubayi and Verstraete (2005) for all n > 3r/2.