19 problems

Let $[n]$ be the set of integers from $1$ to $n$, and let ${[n] \choose k}$ denote the collection of all $k$-element subsets of $[n]$. For nonnegative integers $k<n$, a subset…

Let $[n]=\{1,\ldots,n\}$, and let ${\Delta}^{Sep}_{k,n}$ be the simplicial complex whose vertices are the $k$-subsets of $[n]$, with faces given by collections of pairwise weakly s…

Let $\Bbb{G}_{k,n}(\Bbb{C})$ be the Grassmannian of $k$-subspaces in $\Bbb{C}^n$. A point is positive when it has a matrix representative whose Plücker coordinates are positive rea…

Let $\operatorname{W}(k,n)$ be the set of maximal collections of pairwise weakly separated $k$-subsets of $[1\dots n]$, and let a $(2,4)$-move replace one of the two indicated memb…

Let $[1\dots n]=\{1,\dots,n\}$, and let a collection of $k$-subsets be pairwise weakly separated when each pair satisfies the weak separation relation used in the source. Let…

Let $J,J'$ be $k$-element subsets of $[n]$, and let $L(M_J)$ and $L(M_{J'})$ be the corresponding simple $U_q(\widehat{\mathfrak{sl}_k})$-modules. Weak-separation tensor-product co…

Linear-ceiling formula conjecture. There exist real numbers $a_k$ and $b_k$ such that

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…

For odd $r$ and $n>r$, let ${\bf W}_{n,r}$ denote the poset of weakly $r$-separated collections in $2^{[n]}$, ordered by inclusion. A poset is pure when all its maximal elements ha…

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

Purity conjecture. All maximal by inclusion collections of pairwise weakly separated subsets of $[n]$ have the same cardinality.

Let $[n]$ be the set of integers from $1$ to $n$, and let ${[n] \choose k}$ denote the collection of all $k$-element subsets of $[n]$. For nonnegative integers $k<n$, a subset…

Pairwise smallest-minors conjecture. There exists an arrangement of smallest minors $S\subset\binom{[n]}{k}$ containing $I$ and $J$ if and only if either the pair is 1-interlaced,…

Maximal-size arrangement conjecture. Any arrangement of smallest minors in $Gr^+(k,n)$ contains at most

For $2\leq k\leq n-2$, let $\widetilde{\Delta}^{Sep}_{k,n}$ be the reduced complex obtained from the weak separation complex of weakly separated $k$-subsets of $[n]$ by removing it…

Let $[n]=\{1,2,\ldots,n\}$, and let $\mathcal{C}$ be a collection of subsets of $[n]$ that are pairwise weakly separated, meaning that for every two members $I,J\in\mathcal{C}$, ei…

Let $[m]=\{1,\ldots,m\}$, and let ${\bf m}^{(ws)}_{k \leq l}$ be the compatibility complex whose vertices are the subsets of $[m]$ having at least $k$ and at most $l$ elements, wit…