266 problems

Affine-plane representability problem. Determine whether

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…

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…

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

Let $k\geq 2$. A flag sphere is a simplicial sphere with no missing faces of dimension larger than one, and $g_0,g_1,\ldots,g_k$ are its $g$-numbers. Charney–Davis conjecture. Ever…

Let $\Delta$ be a flag simplicial sphere of dimension $d-1$, and let … be the coefficients in its gamma expansion. Nevo–Petersen conjecture. There exists a simplicial complex…

Let $H_s(n,d)$ denote the maximum diameter of a $d$-dimensional simplicial complex on $n$ vertices. The simplicial-complex diameter conjecture. For every $d\geq 3$, there is a thre…

Hexagonal line tiling shellability conjecture. For $k=4,5,$ and $6$, the $k$-cut complex of the hexagonal line tiling is shellable.

Let $G$ be a finite group and let $p$ be a prime dividing $|G|$. Let ${\mathcal{A}}_p(G)$ be the poset of non-trivial elementary abelian $p$-subgroups of $G$, ordered by inclusion,…

Let $\Delta_{n-1}^{(d)}$ denote the $d$-skeleton of the $(n-1)$-dimensional simplex. A pure $d$-dimensional shellable complex on $n$ vertices is extendably shellable if every shell…

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…

Kalai–Sarkaria conjecture. If $K$ admits an embedding into $S^{r}$, then

Let $k\geq 2$, and let $\Delta$ be a neighborly $2k$-sphere with $n$ vertices. Write $m_{k+1}(\Delta)$ for its number of missing faces of dimension $k+1$. Missing-value conjecture.…

Let $\Delta_{d-1}$ be the standard $(d-1)$-simplex, and let $n\Delta_{d-1}$ denote its dilation by $n$. A unit simplex is a simplex of normalized volume one in $n\Delta_{d-1}$, and…

Let $A_S$ be an Artin group, and let $S'ubset S$ be almost spherical. Let $\Delta_{S,S'}$ denote the relative Artin complex spanned by vertices of types indexed by $S'$. Assume tha…

Redundant symmetry-rigidity conjecture. For every such $\Gamma$ and $e$, $(G({\cal S})-e-e^*,*)$ is $\Gamma$-rigid in $\mathbb R^{k+1}$.

Let $T$ be a tree and let $r\geq 1$. The complex $\operatorname{Ind}_r(T)$ is the $r$-independence complex of $T$. Tree higher-independence conjecture. For any tree $T$ and…

Kalai–Sarkaria face-number conjecture. The complex $K$ has at most as many $d$-simplices as $\partial C_{r+1}(n)$, and

Stanley's partitionability conjecture. Every Cohen–Macaulay simplicial complex is partitionable.

Bounded regularity subdivision conjecture. There exist $M_\epsilon$ and $M_\epsilon'$ such that

Lickorish's conjecture. Any simplicial subdivision of a $d$-simplex is collapsible.

Let $t\leq r$ be two natural numbers. A family of faces is $t$-intersecting if every pair of faces in it has intersection of cardinality at least $t$. Let $\Delta$ be a simplicial…

Björner–Swartz conjecture. The $g$-vector of any doubly Cohen–Macaulay complex is an $M$-sequence.

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…