# Conjectures on union-closed families of sets

Christopher Bouchard – GJM, Volume 9, Issue 1 (2024), 51-59.

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- septembre 16, 2024
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A family of sets \mathcal{A} is union-closed if it is finite and nonempty with member sets that are all finite and distinct (at least one of which is nonempty) and it satisfies the property X, Y \in \mathcal{A} \implies X \cup Y \in \mathcal{A}. Let \binom{S}{k} be the set of all k-element subsets of a set S, and let [n]=\{1,2,\cdots,n\} represent \bigcup_{A \in \mathcal{A}}A. Further, let \mathcal{A}_B=\{A\in\mathcal{A} \ | \ A \cap B = B\} and \mathcal{A}_{\underline{B}}=\{A\in\mathcal{A} \ | \ A \cap B = \emptyset\}. We consider, for any union-closed family \mathcal{A}, the class of conjectures \textrm{UC}_x \colon \ \exists B \in \binom{[n]}{n-x+1} \ | \ |\mathcal{A}_B| \geq |\mathcal{A}_{\underline{B}}|, where x \in [n]. The extremal case x=n is equivalent to the union-closed sets conjecture, also known as Frankl’s conjecture, which states that there exists an element of [n] that is in at least \frac{|\mathcal{A}|}{2} member sets of \mathcal{A}. We prove \textrm{UC}_x for x \in [\lceil \frac{n}{3} \rceil + 1], and also investigate two strengthenings of the union-closed sets conjecture.

#### Milestones:

Received: October 28, 2023

Accepted: March 14, 2024

Revised: August 11, 2024

#### Authors:

__Christopher Bouchard__

1971 Western Ave #107,

Albany, NY 12203, USA.