Conjectures on union-closed families of sets

Abstract

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.