5 problems

Jonsson's polytopality conjecture. For every $n\geq 2k+1$, the complex $\overline{\mathcal{A}\hskip-0pt\mathit{ss}}_{k}(n)$ is a polytopal sphere: there is a simplicial polytope of…

For any integer $k 1$, let a $(k,\nu)$-tree be a maximal subset of $A_\nu$ without $k+1$ pairwise $\nu$-incompatible elements, and let the $(k,\nu)$-Tamari complex be the simplicia…

Fan realization conjecture. Every such choice of ordered parameters gives a configuration in the moment curve that realizes $\overline{\mathcal{A}\hskip-0pt\mathit{ss}}_{2}(n)$ as…

Arbitrary-position rigidity conjecture. Reduced bipartized $2$-triangulations are bases in the bipartite bar-and-joint rigidity matroid of these points.

Rigidity realization conjecture. Reduced bipartized $k$-triangulations of the $n$-gon are bases in the bar-and-joint rigidity matroid of these generic points.