4 problems

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…

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 $f(n,k,l)$ 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…