6 problems

Skeleton-ridge-chordality conjecture. If $\Delta^\vee$ is vertex-decomposable, then $\Delta$ is skeleton-ridge-chordal.

Let $A$ be a vertex decomposable flag simplicial complex, and let $h(A)$ denote its $h$-vector. A face-vector conjecture. The vector $h(A)$ should be the face vector of some flag s…

Let $T$ be a tree, and for a positive integer $r$ let $\mathrm{Ind}_r(T)$ be the complex of $r$-independent sets of $T$, meaning that every connected component of the induced subgr…

Let $\mathcal{S}(Q,\pi)$ be a spherical subword complex of infinite type. Strong vertex-decomposability conjecture. The complex $\mathcal{S}(Q,\pi)$ is strongly vertex decomposable…

Let $\mathcal{S}(Q,\pi)$ be a spherical subword complex of infinite type, and let $\{i\}\in\mathcal{S}(Q,\pi)$ be a vertex. The infinite-type deletion conjecture. The deletion … is…

Let $f$ be a finite sequence of positive integers. A flag complex is a simplicial complex whose minimal non-faces are edges; its face vector records the numbers of faces in each di…