Closed Intersecting Families of finite sets and their applications

TL;DR

This paper introduces closed intersecting families of finite sets, shows they can be embedded into maximal intersecting families, and uses this to disprove some conjectures while providing simpler constructions for large families.

Contribution

It defines closed intersecting families, demonstrates their embedding into maximal families, and offers new constructions and counterexamples related to longstanding conjectures.

Findings

Closed intersecting families can be embedded into maximal intersecting families.

Counterexamples are provided to two special cases of Frankl et al.'s conjecture.

A simpler construction method for large maximal intersecting families is presented.

Abstract

Paul Erd\H{o}s and L\'aszl\'o Lov\'asz established that any \emph{maximal intersecting family of $k-$sets} has at most $k^{k}$ blocks. They introduced the problem of finding the maximum possible number of blocks in such a family. They also showed that there exists a maximal intersecting family of $k-$sets with approximately $(e-1)k!$ blocks. Later P\'eter Frankl, Katsuhiro Ota and Norihide Tokushige used a remarkable construction to prove the existence of a maximal intersecting family of $k-$sets with at least $(\frac{k}{2})^{k-1}$ blocks. In this article we introduce the notion of a \emph{closed intersecting family of $k-$sets} and show that such a family can always be embedded in a maximal intersecting family of $k-$sets. Using this result we present two examples which disprove two special cases of one of the conjectures of Frankl et al. This article also provides comparatively simpler construction of maximal intersecting families of $k-$sets with at least $(\frac{k}{2})^{k-1}$ blocks.