266 problems

Let $\mathrm{Fw}_n$ denote the boundary framework, let $L_r(n)$ be the vertices of local simplex dimension $r$, and let $(n)$ and $(1^n)$ be the two antenna vertices. Low-dimension…

Antenna-vertex conjecture. For every $n\geq 4$, the only vertices of local simplex dimension $1$ are the two antenna vertices:

Let $L_r(n)$ be the vertices of local simplex dimension $r$ in the partition graph, let $\tau(r)$ be the least $n$ for which $L_r(n)$ is nonempty, and let…

Let $G_n$ be the partition graph, let $K_n$ be its clique complex, let $L_r(n)$ denote the vertices of local simplex dimension $r$, let $\Delta(n)$ be the maximal local simplex dim…

Let ${\mathcal L}$ be a lattice and let $C$ be a crosscut of ${\mathcal L}$. Write $\Gamma(C,{\mathcal L})$ for the crosscut complex, $\operatorname{Bd}(\Gamma(C,{\mathcal L}))$ fo…

Cohen–Macaulayness conjecture. The complex $\Sigma_r(G)$ is Cohen–Macaulay if and only if $\operatorname{Con}_r(G)$ is co-chordal.

Let $G$ be a circle graph, and let $I(G)$ be its independence complex. The size of the input is measured by the number of vertices of $G$. Polynomial-time homotopy-type conjecture.…

Let $\Delta$ be a flag homology sphere. The vectors $g(\Delta)$ and $h(\Delta)$ record the face numbers of simplicial complexes, with the latter specifically recording the face num…

Let $\Delta$ be a connected $(d-1)$-dimensional homology manifold with nonempty orientable boundary, where $d\geq 4$. For each face $F$ of $\Delta$, let…

Multiplicity conjecture. The inequalities

Non-embeddability conjecture. The complex $L$ does not embed in $\mathbb{R}^{2d}$. This generalizes the example discussed in the source and is described there as joint work in prog…

Let $H$ and $K$ be simplicial complexes, and write $H<K$ when $H$ is a minor of $K$, obtained by admissible contractions and deletions. Let $m$ be a nonnegative integer. Minor mono…

Kalai–Kleinschmidt–Lee conjecture. For every $j$, the number of $j$-dimensional empty simplices of $P$ is at most the number of $j$-dimensional empty simplices of…

Let $N$ and $N'$ be matroids, and let $N \rightarrow N'$ be a strong map. Write $IN(N)$ and $IN(N')$ for their independence complexes, and let $(IN(N),IN(N'))$ denote the resulting…

Let $X$ be a doubly CohenMacaulay, or $2$-CM, complex, and let $g(X)$ denote its $g$-vector. An $M$-sequence is the $f$-vector of an order ideal of monomials. Bjrner and Swartz's…

Let $\{X_i\}_{i=1}^\infty$ be a family of finite simplicial complexes. The strong NE-equivalence conjecture. There exists an infinite family of finite simplicial complexes…

Let $X$ and $Y$ be finite simplicial complexes. The NE-equivalence conjecture. There exist finite simplicial complexes $X$ and $Y$ having the same simple homotopy type, such that…

Let $\Delta$ be an $l$-CM simplicial complex of maximal girth. Let $c$ denote the parameter used for its dimension in the paper, and suppose that $\Delta$ is not the $r$-skeleton o…

A finite simplicial complex is the topological space obtained by gluing finitely many simplices along faces. A compact metrizable space is finitely self-similar if it arises from a…

Radicality conjecture. The ideal $I_{\Delta}$ is radical if and only if

Primality and perfection conjecture. If $\Delta$ is any simplicial complex, then

Let $d$ and $q$ be positive integers, and let $\Delta_{(d+1)(q-1)}^{\le d}$ denote the $d$-skeleton of the simplex. A Tverberg partition is a collection of $q$ disjoint faces whose…

Let ${\cal K}$ be a simplicial complex of type $\{3,4\}$, meaning that every $(d-2)$-face is contained in $3$ or $4$ faces of dimension $d-1$. Let $(P_1,\ldots,P_t)$ be the corresp…

Let ${\cal K}$ be a complex of odd dimension. A zigzag in ${\cal K}$ has a signature, recording the numbers of intersections of the zigzag with the relevant types of faces. Zero-si…

A complementary weak pseudomanifold is a weak pseudomanifold whose simplicial complex satisfies complementarity. Let $M$ be a $d$-dimensional complementary weak pseudomanifold with…