9 problems

Let $s$ be a positive integer, and let $L$ be a set of $s$ positive integers. For a positive integer $n$, write $[n]=\{1,\ldots,n\}$, and let $2^{[n]}$ denote the family of all sub…

Balogh–Linz–Patkós' conjecture. There exists a positive integer $n_0=n_0(k,t)$ such that if $n+t$ is odd, $n>n_0$, and $\mathcal{F}\subseteq 2^{[n]}$ is a $t$-intersecting $k$-Sper…

Uniform Sperner-family balance conjecture. If

Let $[n]=\{1,\ldots,n\}$, let $\mathcal{S}\subseteq 2^{[n]}$ be a Sperner family, meaning that no member of $\mathcal{S}$ contains another, and let $h:\mathcal{S}\to 2^{[n]}$ satis…

Let $[n]=\{1,\ldots,n\}$, let $2^{[n]}$ be the family of subsets of $[n]$, and let a set system $\mathcal{F}\subseteq 2^{[n]}$ be s-extremal if it shatters exactly $|\mathcal{F}|$…

Let $p$ and $q$ be different positive parameters, and let a $(p,q)$-tilted Sperner family with patterns on $[n]$ be a family of subsets of $[n]$ satisfying the relevant tilted Sper…

Let $[n]=\{1,\ldots,n\}$, and let $f(n,k,n-l)$ denote the maximum size of an $(n-l)$-trace $k$-Sperner family in $2^{[n]}$. For positive integers $k,l$ with $l<k$, write…

Let $k$ and $n$ be positive integers, and let ${\mathcal F}\subseteq 2^{[n]}$ be a strongly saturating $k$-Sperner family: it contains no $k+1$-chain, and every set outside…

Distance-one extremal conjecture. Every family $\mathcal{A}\subset\mathcal{P}[n]$ that forbids distance one satisfies