Set systems without a 3-simplex

Abstract

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 $2^{n-1}+\left(\genfrac{}{}{0ex}{}{n-1}{0}\right)+\left(\genfrac{}{}{0ex}{}{n-1}{1}\right)+\left(\genfrac{}{}{0ex}{}{n-1}{2}\right)$ for all $n\ge 1$, with equality only achieved by the family of sets either 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.

5 pages