6 problems

Let $n,s$ be integers with $n\geq 2s-1$, let $L=\{\ell_{1},\ell_{2},\ldots,\ell_{s}\}$ be a set of $s$ non-negative integers, and let an $L$-intersecting Sperner system be a Sperne…

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…

Let $n,t,k$ be positive integers, and let ${\mathcal B}(t,k)$ denote the family constructed in the source: it contains the $(n+t-1)/2$-subsets containing $[1,t]$, all sets whose si…

Let $n$ and $t$ be integers with $n+t$ even and $n>t$, and let ${\mathcal F}\subseteq 2^{[n]}$ be a $t$-intersecting family, meaning that $|F\cap F'|\geq t$ for all…

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…

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