Union-Closed vs Upward-Closed Families of Finite Sets

TL;DR

This paper explores the properties of union-closed families of finite sets, establishing bounds on element lengths and the number of joint-irreducible elements using advanced combinatorial techniques.

Contribution

It introduces new bounds on the average element length and the maximum number of joint-irreducible elements in union-closed families, extending previous connections with upward-closed families.

Findings

Established tight lower bounds for average element length.

Proved an upper bound on the number of joint-irreducible elements.

Extended the connection between union-closed and upward-closed families.

Abstract

A finite family $\mathrsfs{F}$ of subsets of a finite set $X$ is union-closed whenever $f,g\in\mathrsfs{F}$ implies $f\cup g\in\mathrsfs{F}$. These families are well known because of Frankl's conjecture. In this paper we developed further the connection between union-closed families and upward-closed families started in Reimer (2003) using rising operators. With these techniques we are able to obtain tight lower bounds to the average of the length of the elements of $\mathrsfs{F}$ and to prove that the number of joint-irreducible elements of $\mathrsfs{F}$ can not exceed $2{n\choose \lfloor n/2\rfloor}+{n\choose \lfloor n/2\rfloor+1}$ where $|X| = n$.