Set systems without a 3-simplex

A 3-simplex is a collection of four sets A 1 , … , A 4 with empty intersection such that any three of them have nonempty intersection. We show that the maximum size of a set system on n elements without a 3 -simplex is 2 n − 1 + ( n − 1 0 ) + ( n − 1 1 ) + ( n − 1 2 ) for all n ≥ 1 , with equality only achieved by the family of sets containing a given element or of size at most 2 . This extends a result of Keevash and Mubayi, who showed the conclusion for n sufficiently large.