An intersection theorem for four sets

Fix integers n, r ≥ 4 and let F denote a family of r-sets of an n-element set. Suppose that for every four distinct A,B,C,D ∈ F with |A∪B ∪C ∪D| ≤ 2r, we have A∩B ∩C ∩D 6= ∅. We prove that for n sufficiently large, |F| ≤ ( n−1 r−1 ) , with equality only if ⋂ F∈F F 6= ∅. This is closely related to a problem of Katona and a result of Frankl and Füredi [10], who proved a similar statement for three sets. It has been conjectured by the author [18] that the same result holds for d sets (instead of just four), where d ≤ r, and for all n ≥ dr/(d− 1). This exact result is obtained by first proving a stability result, namely that if |F| is close to ( n−1 r−1 ) then F is close to satisfying ⋂ F∈F F 6= ∅. The stability theorem is analogous to, and motivated by the fundamental result of Erdős and Simonovits for graphs.