The number of abundant elements in union-closed families without small sets

TL;DR

This paper investigates the number of abundant elements in union-closed families without small sets, providing new bounds and constructions that relate to longstanding conjectures in combinatorics.

Contribution

It establishes new lower bounds on the number of abundant elements and constructs examples that meet these bounds, advancing understanding of union-closed families.

Findings

Proves that such families have at least k abundant elements if k ≥ n - 3.

Constructs families with exactly k - 1 abundant elements for certain parameters.

Shows that the number of abundant elements is always at least min{n, 2k - n + 1}.

Abstract

We let $\mathcal{F}$ be a finite family of sets closed under taking unions and $\emptyset \not \in \mathcal{F}$, and call an element abundant if it belongs to more than half of the sets of $\mathcal{F}$. In this notation, the classical Frankl's conjecture (1979) asserts that $\mathcal{F}$ has an abundant element. As possible strengthenings, Poonen (1992) conjectured that if $\mathcal{F}$ has precisely one abundant element, then this element belongs to each set of $\mathcal{F}$, and Cui and Hu (2019) investigated whether $\mathcal{F}$ has at least $k$ abundant elements if a smallest set of $\mathcal{F}$ is of size at least $k$. Cui and Hu conjectured that this holds for $k = 2$ and asked whether this also holds for the cases $k = 3$ and $k > \frac{n}{2}$ where $n$ is the size of the largest set of $\mathcal{F}$. We show that $\mathcal{F}$ has at least $k$ abundant elements if $k \geq n - 3$, and that $\mathcal{F}$ has at least $k - 1$ abundant elements if $k = n - 4$, and we construct a union-closed family with precisely $k - 1$ abundant elements for every $k$ and $n$ satisfying $n - 4 \geq k \geq 3$ and $n \geq 9$ (and for $k = 3$ and $n = 8$). We also note that $\mathcal{F}$ always has at least $\min \{ n, 2k - n + 1 \}$ abundant elements. On the other hand, we construct a union-closed family with precisely two abundant elements for every $k$ and $n$ satisfying $n \geq \max \{ 3, 5k-4 \}$. Lastly, we show that Cui and Hu's conjecture for $k = 2$ stands between Frankl's conjecture and Poonen's conjecture.