Non-trivial 3-wise intersecting uniform families

TL;DR

This paper investigates the maximum size of 3-wise intersecting families of k-element subsets from an n-element set, providing exact or asymptotic bounds and revealing new structural insights.

Contribution

It determines the maximum size of non-trivial 3-wise intersecting uniform families, extending understanding beyond trivial or pairwise intersecting cases.

Findings

Exact maximum size for certain parameters

Asymptotic bounds for large n

Structural characterization of extremal families

Abstract

A family of $k$-element subsets of an $n$-element set is called 3-wise intersecting if any three members in the family have non-empty intersection. We determine the maximum size of such families exactly or asymptotically. One of our results shows that for every $\epsilon>0$ there exists $n_0$ such that if $n>n_0$ and $\frac25+\epsilon<\frac kn<\frac 12-\epsilon$ then the maximum size is $4\binom{n-4}{k-3}+\binom{n-4}{k-4}$.