Diversity of uniform intersecting families

Andrey Kupavskii

Published 2017-09-08Version 1

A family $\mathcal f\subset 2^{[n]}$ is called {\it intersecting}, if any two of its sets intersect. Given an intersecting family, its {\it diversity} is the number of sets not passing through the most popular element of the ground set. 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 $k$.