14 problems

Simonyi's sandglass conjecture. Every recovering pair $(\mathcal{A},\mathcal{B})$ over $[n]$ satisfies

Let $\mathcal{F}\subset 2^{[n]}$ be a family of subsets of $[n]$, and let $\ell$ be a positive integer. Suppose that, for some integer $k=k(\ell)$, the intersection of any $k$ dist…

Moss–Pedersen conjecture. For all sufficiently large $n$, there exists a family $\mathcal F\subseteq\binom{[2n]}{n}$ such that

Erdős–Kleitman matching conjecture. Suppose that $s\ge 2$, $m\ge 1$, and $n=sm+s-\ell$ for some integer $0<\ell\leq\lceil s/2\rceil$. Then

The linear-or-bounded conjecture. For every finite poset $\mathcal P$, the induced saturated number $\operatorname{sat}^*(n,\mathcal P)$ is either bounded, or at least linear in…

Let $[n]=\{1,\ldots,n\}$, let $2^{[n]}$ denote the power set of $[n]$, and let an IU-family be a family $\mathcal{G}\subseteq 2^{[n]}$ such that … and … Write $\beta(\mathcal{G})$…

Let $G_4=(A,B,E)$ be the bipartite incidence graph of the complete graph $K_4$, with $A=E(K_4)$, $B=V(K_4)$, and $(e,v)\in E$ exactly when $v\in e$. Let $P_{G_4,A}$ and $P_{G_4,B}$…

Let $G$ be a bipartite graph that is not a matching. For a poset $P$, let $\mathop{}\!\mathrm{La}_{sym}(n,P)$ denote the maximum size of a $P$-free complement-closed family of subs…

Dense minimizer conjecture. There exists a family

Sparse minimizer conjecture. There exists a minimizer $\mathcal{F}$ such that

Let $P$ be a poset, and call a family $\mathcal F\subseteq 2^{[n]}$ $P$-free if it contains no copy of $P$. Let $La(n,P)$ denote the maximum size of a $P$-free family in $2^{[n]}$.…

Borg's product conjecture. The product

Borg's stronger sum conjecture. (i) If

Borg's sum conjecture. If $k\geq n+1$, then