On regular 3-wise intersecting families

TL;DR

This paper proves that any regular, increasing, 3-wise intersecting family of subsets of a finite set has size negligible compared to the power set, extending previous results to broader conditions.

Contribution

It generalizes earlier theorems by showing regular, increasing 3-wise intersecting families are small, relaxing the transitivity condition.

Findings

Regular, increasing 3-wise intersecting families are of size o(2^n)

Extends previous results by removing the transitivity requirement

Addresses a question posed by Cameron, Frankl, and Kantor in 1989

Abstract

Ellis and the third author showed, verifying a conjecture of Frankl, that any $3$-wise intersecting family of subsets of $\{1,2,\dots,n\}$ admitting a transitive automorphism group has cardinality $o(2^n)$, while a construction of Frankl demonstrates that the same conclusion need not hold under the weaker constraint of being regular. Answering a question of Cameron, Frankl and Kantor from 1989, we show that the restriction of admitting a transitive automorphism group may be relaxed significantly: we prove that any $3$-wise intersecting family of subsets of $\{1,2,\dots,n\}$ that is regular and increasing has cardinality $o(2^n)$.