NoteSet systems without a 3-simplex

A 3-simplex is a collection of four sets A1,…,A4 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 2n−1+(n−10)+(n−11)+(n−12) 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.

► We consider set systems on n elements without a 3-simplex. ► We determine the maximum size of such systems in terms of n. ► Our results also yield the unique extremal family for all n.