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 Erdős-Simonovits stability theorem. A triangle is a family of three sets A,B,C such that A∩B, B ∩C, C ∩A are each nonempty, and A∩B∩C = ∅. We prove the following new theorem about the stability of triangle-free set systems. Fix r ≥ 3. For every δ > 0, there exist > 0 and n0 = n0( , r) such that the following holds for all n > n0: if |X| = n and G is a triangle-free family of r-sets of X containing at least (1 − ) (n−1 r−1 ) members, then there exists an (n− 1)-set S ⊂ 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 Erdős (open since 1971) that was only recently settled by Mubayi and Verstraëte (2005) for all n ≥ 3r/2.