Set systems without a 3-simplex

2010/10/25 by Michael E. Picollelli, Picollelli, Michael E.
Computer Science · Mathematics · #Advanced Graph Theory Research #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #math.CO

paper · pdf · doi:10.48550/arxiv.1010.5206

5 pages

arxiv created 2010/10/25 · openalex publication_date 2010/10/25 · arxiv updated 2010/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 2^n-1 + \binomn-10 + \binomn-11 + \binomn-12 for all n ≥ 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.

Related

• Non-trivial 3-wise intersecting uniform families
• Around the Complete Intersection Theorem
• New results on simplex-clusters in set systems
• Families of sets with no matchings of sizes 3 and 4
• Disjoint pairs in set systems with restricted intersection