Diversity of uniform intersecting families

TL;DR

This paper proves a conjecture by Frankl regarding the maximum diversity of intersecting families of k-subsets of an n-set for large n, and discusses related cases where the conjecture does not hold.

Contribution

It confirms Frankl's conjecture for large n relative to k and explores the limitations of natural generalizations for smaller n.

Findings

Proved Frankl's conjecture for n ≥ ck

Identified cases where the generalization does not hold

Analyzed the behavior of intersecting families for different n ranges

Abstract

A family $\mathcal F\subset 2^{[n]}$ is called intersecting if any two of its sets intersect. Given an intersecting family, its diversity is the number of sets not passing through the most popular element of the ground set. Peter Frankl made the following conjecture: for $n> 3k>0$ any intersecting family $\mathcal F\subset {[n]\choose k}$ has diversity at most ${n-3\choose k-2}$. This is tight for the following "two out of three" family: $\{F\in {[n]\choose k}: |F\cap [3]|\ge 2\}$. In this note, we prove this conjecture for $n\ge ck$, where $c$ is a constant independent of $n$ and $k$. In the last section, we discuss the case $2k<n<3k$ and show that one natural generalization of Frankl's conjecture does not hold.