458 problems

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, and let $\Gamma$ be a simplicial complex whose $f$-vector is $\gamma(\Delta)$. Nevo–Petersen balanced conjecture. The complex $\Gamma$ can…

Let $S_{d,n}$ denote the $d$-skeleton of the $(n-1)$-simplex. A pure simplicial complex is extendably shellable if every shelling of it can be extended to a shelling of the whole c…

Iwase–Sakai conjecture.

Let $M$ be a matroid, and let … is the vector recording the face-enumeration data of this simplicial complex. A finite sequence is a pure ${\mathcal O}$-sequence if it is the $f$-v…

Let $R$ be a Euclidean or local ring, and let $\operatorname{CB}(R^n)$ be the simplicial complex whose vertices are the proper nonzero direct summands of $R^n$, with simplices give…

Let $\Delta$ be a simplicial $(d-1)$-sphere. The $g$-numbers are certain alternating weighted sums of the $f$-numbers. McMullen–Walkup conjecture. For any $k\leq d/2$, one has … Th…

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

Let $\Delta$ be a Cohen–Macaulay simplicial complex, and let its $h$-vector provide the prescribed ranks. A Boolean interval in the face poset is an interval isomorphic to a Boolea…

Duval–Reiner conjecture. For every $q$-uniform family $F$ and every $r\ge1$,

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…

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…

Let $k\geq 1$ be an integer, and let $\mathcal{K}_k$ be the simplicial complex whose facets are maximal “$k$-crossing-free” sets of diagonals of a convex polygon, called multi-tria…

Let $P$ be a simple polytope, and let $P^*$ be its polar dual. A Perles piece is a connected simplicial complex such that every facet contains exactly one ridge in the boundary of…

Bjorner--Swartz conjecture. Every doubly Cohen--Macaulay simplicial complex has the dual WLP.

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

Alexander's conjecture. If $K$ and $L$ have a common subdivision, then they have a common stellar subdivision.

Let $p(t)$ be a polynomial with positive integer coefficients and constant term equal to $1$. Bell–Skandera conjecture. The polynomial $p(t)$ is the $f$-polynomial of a simplicial…

Let $\Omega_n=X({\tt K}_n)$ be the cycle-free chessboard complex, and let … Here, a space is $\mu_n$-connected when its homotopy groups vanish through dimension $\mu_n$. Best-conne…

Let $\Delta$ be a balanced $d$-sphere with coloring $\rho: V \to [d+1]$, let $R$ be the polynomial ring whose variables $x_v$ have color $\rho(v)$, and let … Set…

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…

Day–Putman's conjecture. The complex of partial bases $\mathcal{B}_n$ is $(n-2)$-connected.

For a fixed $d$-complex $S$, let $\mathcal{H}(S)$ be the collection of homeomorphs of $S$, and define $\lambda_d$ as the minimum exponent such that … for every fixed $d$-complex…

Let $K$ be a simplicial complex, and let $(\underline{X},\underline{A})^K$ denote the associated polyhedral product. Bahri, Bendersky, Cohen, and Gitler constructed an integral dec…