6 problems

For positive integers $n$ and $d$, let $\mathbf{SQ}(n,d)$ be the digraph whose vertices are symmetric cubillages of the cyclic zonotope $Z([n],d)$ and whose directed edges are the…

Let $n=2m$ or $n=2m+1$, and let $\mathbf{C}(m,d)$ and $\mathbf{B}(m,d)$ be the posets of symmetric cubillages in the cyclic zonotope $Z(n,d)$. Denote their standard and anti-standa…

Size and representability conjecture. The maximal size of a weakly $r$-separated collection $W\subseteq 2^{[n]}$ without double $r$-combs is $s_{n,r}$, and every such collection wi…

Let $r$ be even, let $Q$ be an arbitrary cubillage on $Z(n,r+2)$, and let $M$ be an e-membrane of $Q$. E-membrane weak-separation conjecture. The vertex set $V(M)$ is a weakly $r$-…

Let $r$ be even, let $Q$ be a cubillage on $Z(n,r+2)$, and let $M$ be an e-membrane, meaning an e-membrane of $Q$. Let property (P) mean that $M$ contains no double $r$-comb. Doubl…

Representability conjecture. Every maximal by size weakly $r$-separated collection ${\cal W}\subseteq 2^{[n]}$ is representable. This would characterize all maximal-size weakly $r$…