11 problems

Gerbner et al.'s conjecture. If $n$ is sufficiently large compared to $k$, then the minimum size of a saturated $k$-Sperner system is

Let $[n]=\{1,2,\ldots,n\}$, and let $\Pi_n$ be the refinement order on the set of partitions of $[n]$. An antichain is a collection of pairwise incomparable elements of this poset,…

Uniqueness conjecture. If $2\mid nd$, this set is unique, while if $2\nmid nd$, it is one of exactly two unique largest sets; the other is

Let $n,d,k$ be the parameters in the paper, let $d>2$, and let $A\subseteq\{0,1,\dots,d\}^n$ satisfy the required property: there are no distinct $x,y\in A$ with $x_i\leq y_i$ for…

Let $\Delta_{k,q}$ be the set of integer vectors in $\mathbb{Z}^k$ with non-negative entries summing to $q$, and consider a Sperner-admissible labeling of a regular simplicial subd…

Let $[n]=\{1,\ldots,n\}$, let $2^{[n]}$ be its power set, and let $G_{\mathcal{F}}$ be the graph whose vertex set is a family $\mathcal{F}\subseteq 2^{[n]}$, with two vertices adja…

Higher-order sparse Sperner conjecture. Given $k\in\mathbb N$, there exists a function $\beta^*(n)=\tilde\Theta(1/n^k)$ such that if $\beta\geq\beta^*(n)$, then almost all antichai…

Sparse Sperner conjecture. There exists a function $\beta^*(n)=\tilde\Theta(1/n)$ such that if $\beta\geq\beta^*(n)$, then almost all antichains of size $\beta N$ in $B_n$ are cont…

Let $[s]=\{1,2,\ldots,s\}$, and let $\mathcal{F}\subseteq 2^{[n]}$ be a set system. For distinct $A,B\in\mathcal{F}$, define the skew distance by … The system $\mathcal{F}$ is…

Let $f(n,k,l)$ be the maximum size of an $l$-trace $k$-Sperner family ${\mathcal F}\subseteq 2^{[n]}$. For a family, being $l$-trace $k$-Sperner means that the trace on every $l$-s…

Let $k$ and $l'$ be positive integers with $l'<k$. An $(n-l')$-trace $k$-Sperner family is a family ${\mathcal F}\subseteq 2^{[n]}$ whose trace on every $(n-l')$-set is $k$-Sperner…