34 problems

Let $\mathcal{F}$ be a finite union-closed family of sets, and let $M$ be its largest set. Assume that for every pair of elements $a,b\in M$, the family contains a set $A$ with…

Let $\mathcal{F}$ be a finite union-closed family of sets with $\emptyset \notin\mathcal{F}$. Elements $a$ and $b$ are twins if every set $A\in\mathcal{F}$ satisfies…

Minimal-generator extension conjecture. If $\mathcal{B}$ is not $FC$, then $\mathcal{B}^{\prime}$ is not $FC$ either. This conjecture proposes that replacing a minimal generator by…

For integers $k\geqslant 2$ and $n$, let $FC(k,n)$ denote the threshold quantity for $k$-uniform families on an $n$-element ground set as defined in the paper, and let $\Theta$ hav…

For integers $n\geqslant 4$, let $FC(3,n)$ denote the threshold quantity for 3-uniform families on an $n$-element ground set as defined in the paper. Morris's conjecture. … The con…

Nagel's conjecture. There are $k$ elements, each belonging to at least

Let $\mathcal F$ be a union-closed set family, let $f_k(\mathcal F)$ denote its $k$th-highest element frequency, and fix $k\in\mathbb N$. Asymptotic half-frequency conjecture. As…

Let $G$ be a bipartite graph, and call a vertex pendant when it has degree one. Frankl's pendant-vertex conjecture. Every bipartite graph with at least one pendant vertex satisfies…

Let $G$ be a bipartite graph, and call a vertex pendant when it has degree one. Frankl's one-pendant-vertex conjecture. Every bipartite graph with exactly one pendant vertex satisf…

Jiang's conjecture. For every positive integer $k\leq n$, there exists a set $S\subseteq U(\mathcal{F})$ with $|S|=k$ that is contained in at least

Let $\mathcal{F}$ be a finite normalized family, meaning a normalized union-closed family in the terminology of the source, and let $J(\mathcal{F})$ denote its family of irreducibl…

Let $\mathcal{F}$ be a union-closed family of sets, and suppose that a most frequent element of $U(\mathcal{F})$ belongs to exactly half the sets in $\mathcal{F}$. Exact-half conje…

Let $\mathcal{F}$ be a union-closed family of sets. A power set is a family of the form $\mathcal{P}(X)$ for some set $X$. Poonen's conjecture. Unless $\mathcal{F}$ is a power set,…

Let $\mathcal{F}$ be a finite family of sets that is union-closed, meaning that the union of any two members belongs to $\mathcal{F}$. Suppose that every set in $\mathcal{F}$ has s…

Lexicographic initial-segment conjecture. For fixed $n>k\geq3$, if $[S_m]$ is an FC-family for some positive integer $m$ and has universe size $n$, then

Zeon formulation of Frankl's conjecture. There exists $i\in[n]$ such that

Positive average overlap density conjecture. There exists an absolute positive constant $c>0$ such that, for every $n\in\mathbb{N}$ and every such $\mathcal{F}$, the average overla…

Extremal formulation of Frankl's conjecture. For every $a,n\in\mathbb{N}^+$,

Let $[n]$ be a finite universe, let $\mathcal F\subseteq 2^{[n]}$ be a union-closed family, and define its closure by … Suppose that $\overline{\mathcal F}$ satisfies Frankl's unio…

Let $\widetilde{\mathfrak{A}}_{k|\ell}$ be the class of weakly $k|\ell$-separated union-closed families, and let $\widetilde{\mathfrak{c}}_{k|\ell}$ be the corresponding supremal p…

Let $\mathfrak{A}_{k|\ell}$ be the class of union-closed families used in the paper, and let $\mathfrak{c}_{k|\ell}$ denote the corresponding supremal proportion of members meeting…

Let $\mathcal{A}$ be a finite union-closed collection of sets containing at least one non-empty set, and let $|\mathcal{A}|$ denote its number of sets. The $\varepsilon$-union-clos…

Let $[n]=\{1,\ldots,n\}$, let ${\mathcal A}\subseteq 2^{[n]}$, and suppose ${\mathcal A}$ satisfies Condition 1: there is a filter ${\mathcal F}\subseteq 2^{[n]}$ and a bijection…

Let $[n]=\{1,\ldots,n\}$ and let $U(\mathcal{S})$ denote the union of the sets in $\mathcal{S}$. A minimal generator of a union-closed family $\mathcal{F}$ is a family whose genera…

Let $f(n,a)$ denote the optimization quantity defined in the paper for union-closed families on $n$ ground-set elements with maximum element frequency at most $a$. $f$-conjecture.…