9 problems

Let $[n]=\{1,\ldots,n\}$, and let $\mathcal{F}\subseteq 2^{[n]}$ be a non-empty union-closed family, meaning that $A\cup B\in\mathcal{F}$ for all $A,B\in\mathcal{F}$. Let…

Let $M=(m_{ij})$ be a non-zero binary matrix of size $n\times m$ with distinct rows. For binary rows, write $M_{i-}\lor M_{j-}$ for their coordinatewise logical disjunction, and de…

Let $A$ and $B$ be independent identically distributed samples from a distribution supported on a family of subsets of $[n]$. Write $H(A)$ for the entropy of $A$, and…

Let $\mathcal{N}\subset\mathbb{N}^+$ be a finite GCD-closed set, meaning that $\operatorname{gcd}(m,n)\in\mathcal{N}$ whenever $m,n\in\mathcal{N}$, and suppose that $\mathcal{N}$ c…

Let $\mathcal{N}\subset\mathbb{N}^+$ be a finite LCM-closed set, meaning that $\operatorname{lcm}(m,n)\in\mathcal{N}$ whenever $m,n\in\mathcal{N}$, and suppose that $\mathcal{N}$ c…

Let $F$ be a reduced union-closed matrix, let $F^T$ be its transpose, and let $\overline{F}$ be the characteristic matrix of the complements of the represented sets. Write…

Let $\mathcal{F}$ be a reduced union-closed family on a set $S$. Union-intersection formulation of the strong union-closed sets conjecture. The average cardinality of unions of ord…

Let $\mathcal{F}$ be a reduced union-closed family on a set $S$. Strong union-closed sets conjecture. The average cardinality of members of $\mathcal{F}$ is not greater than twice…

Let $F \in \mathbb{Q}^{m \times n}$ be a reduced intersection-closed matrix, and let $e_k$ be the $k$-th unit coordinate vector. Write $\Sigma X$ for the sum of all entries of a ma…