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 $$\mathcal{A}<em>B={A\in\mathcal{A} \ | \ A \cap B = B}$$ and $$\mathcal{A}</em>{\underline{B}}={A\in\mathcal{A} \ | \ A \cap B = \emptyset}$$. We consider, for any union-closed family $\mathcal{A}$, the class of conjectures $$\textrm{UC}<em>x \colon \ \exists B \in \binom{[n]}{n-x+1} \ | \ |\mathcal{A}_B| \geq |\mathcal{A}</em>{\underline{B}}|$$, 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.