Intersecting families with covering number $3$

TL;DR

This paper determines the largest intersecting families of k-element subsets with covering number 3 for large enough n and k, extending Frankl's 1978 results and settling the problem for certain parameter ranges.

Contribution

It proves that Frankl's extremal family remains optimal for all sufficiently large k and n, effectively resolving the longstanding problem.

Findings

Frankl's family is extremal for n > 2k and k ≥ 100

The result holds for all n ≥ 2k and large k

The paper settles the problem for the specified parameter ranges

Abstract

A covering number of a family is the size of the smallest set that intersects all sets from the family. In 1978 Frankl determined for $n\ge n_0(k)$ the largest intersecting family of $k$-element subsets of $[n]$ with covering number $3$. In this paper, we essentially settle this problem, showing that the same family is extremal for any $k\ge 100$ and $n>2k$.