48 problems

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

Let $G$ be a bipartite graph with at least one edge, and let a maximal stable set be a vertex subset containing no adjacent vertices and maximal with respect to inclusion. Graph fo…

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 union-closed family of finite sets. Gilmer's proposed bound. Similar information-theoretic methods should lead to the existence of an element that bel…

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

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…

Let $[n]=\{1,\ldots,n\}$, and let $2^{[n]}$ denote the family of subsets of $[n]$. A family $\mathcal F\subseteq 2^{[n]}$ is union-closed if the union of any two members of…

Let $n\geq 2$, let $M_n=\{1,2,\ldots,n\}$, and let $\mathcal{F}\subset 2^{M_n}$ be a finite union-closed family with $\bigcup_{A\in\mathcal{F}}A=M_n$. For $i\in M_n$, write…

Let $M_n=\{1,2,\ldots,n\}$ and let $\mathcal{F}\subset 2^{M_n}$ be a union-closed family with $\bigcup_{A\in\mathcal{F}}A=M_n$. Define … where…

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 $\mathcal{A}$ be a finite union-closed family of sets with $\emptyset\notin\mathcal{A}$, and let $\mathcal{P}_1,\ldots,\mathcal{P}_w$ be the classes produced by the paper's par…

Let $\mathcal{A}$ be a finite union-closed family of sets, and let $\mathcal{P}=\{\mathcal{P}_1,\ldots,\mathcal{P}_w\}$ be the partition constructed by the paper's algorithm, with…

Let $\mathcal{A}\neq\{\emptyset\}$ be a finite union-closed family of sets with base set $[n]=\bigcup_{A\in\mathcal{A}}A$. For $B\subseteq[n]$, define … For a positive integer $k$,…

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…

Let $A$ and $B$ be independent identically distributed samples from a distribution over a family of subsets of $\{1,2,\ldots,n\}$. Assume that…

Let $n\in\mathbb{N}$, let $[n]=\{1,\dots,n\}$, and let $\mathcal{P}(n)=\mathcal{P}([n])$ be the power set of $[n]$. A family $\mathcal{F}\subseteq\mathcal{P}(n)$ is nontrivial when…

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…

Union-closed sets conjecture. If $\mathcal{A}\subset 2^{[n]}$ is union-closed, then there is $k\in[n]$ such that