6 problems

Let $\operatorname{W}(k,n)$ be the set of maximal collections of pairwise weakly separated $k$-subsets of $[1\dots n]$, and let $\Delta^I$ denote the Plücker coordinate indexed by…

Let $\operatorname{Gr}(k,n)$ be a Grassmannian, and let a primitive ratio be … where $i<i+1<j<j+1$ are cyclically ordered and $S$ is a $(k-2)$-element subset of $[n]$ disjoint from…

Let $G(n-d,n)$ denote the Grassmannian of $(n-d)$-dimensional subspaces of an $n$-dimensional space, and let $\bm{p}$ be a Plücker-coordinate vector in $\bigcup\limits_n G(n-d,n)$.…

Let $V$ and $W$ be matrices whose stacked matrix is $U:=\operatorname{stack}(V,W)\in\operatorname{Mat}^{\circ}(k+\ell,n)$, and let the stacked twist map satisfy…

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