Diversity of uniform intersecting families

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 $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ has diversity at most $\left(\genfrac{}{}{0ex}{}{n-3}{k-2}\right)$. This is tight for the following "two out of three" family: $\{F\in (\genfrac{}{}{0ex}{}{[n]}{k}):|F\cap [3]|\ge 2\}$. 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.