Set systems without a 3-simplex

TL;DR

This paper determines the maximum size of a set system on n elements that avoids a 3-simplex, extending previous results to all n and characterizing the extremal families.

Contribution

It provides an exact formula for the maximum size of 3-simplex-free set systems for all n and characterizes the extremal families achieving this maximum.

Findings

Maximum size formula for 3-simplex-free set systems.

Characterization of extremal families containing a fixed element or small sets.

Extension of previous results to all n, not just large n.

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 $2^{n-1} + \binom{n-1}{0} + \binom{n-1}{1} + \binom{n-1}{2}$ 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.