11 problems

Let $C$ be a simplicial complex, and let $|C|$ denote its underlying space. The space $|C|$ is star-shaped if it is star-shaped in the relevant Euclidean space. Goodrick's conjectu…

Let a good cover in $\mathbb{R}^d$ be a collection of open sets such that every subcollection has intersection either empty or homeomorphic to an open $d$-ball. The nerve of a coll…

Let $\mathbb{I}_n$ denote the $n$-dimensional hypercube graph, and let $\mathcal{VR}(\mathbb{I}_n;r)$ be its Vietoris–Rips complex at scale $r$. Let the collapsibility number of a…

Collapsible-to-Leray tolerance conjecture. Then $\mathcal{T}_{t}(K)$ is $(\eta(d+1,t+1)-1)$-Leray.

Link collapsibility conjecture. If $A$ is an independent set of size $n-1$ in $G$, then

Collapsibility conjecture. If $G$ has maximum degree at most $\Delta$, then

Let $d\geq 2$, and consider $d$-dimensional hypertrees with complete $(d-1)$-skeleton. A $d$-dimensional hypertree is a $d$-dimensional $\mathbb{Q}$-acyclic complex on a vertex set…

Let an $athfrak{I}$-contractible graph be a graph reducible to a single vertex by $athfrak{I}$-contractible transformations. An expansion is the inverse of a collapse, and a graph…

Let $athfrak{I}$-contractible graphs be graphs reducible to a single vertex by strong $athfrak{I}$-contractible transformations, and let collapsible graphs be graphs whose clique c…

Let $P$ be a compact $n$-polyhedron. A free deformation retraction of $P$ onto a subspace $Q$ is a homotopy $h_t:P\to P$ with $h_0=\operatorname{id}$, $h_1$ a retraction onto $Q$,…

Let ${\cal F}_{n,d}$ denote the family of $d$-dimensional random simplicial complexes on $n$ vertices, and let $\gamma_d$ be the constant defined in the paper. Threshold conjecture…