23 problems

NP-hardness conjecture. It is NP-hard to decide if a given $N$-bit vector $f=(1,f_0,\ldots,f_{d-1})$ of positive integers is the $f$-vector of a $d$-polytope.

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…

Kühn­el's componentwise minimality conjecture. The $f$-vector

$L(3,1)$ $f$-vector conjecture. The $f$-vectors of triangulations of the lens space $L(3,1)$ are characterized by

3-torus $f$-vector conjecture. The $f$-vectors of triangulations of the $3$-torus $T^3$ are characterized by

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 $P$ be a convex $d$-polytope with $f$-vector $(f_0,f_1,\ldots,f_{d-1})$, where $f_k$ is the number of $k$-dimensional faces. The face-number lower-bound conjecture. For every…

Let $P$ be a convex $d$-polytope with $f$-vector $(f_0,f_1,\ldots,f_{d-1})$, where $f_k$ denotes the number of $k$-dimensional faces. Bjrner's quarter-monotonicity conjecture. The…

Let $P$ be a convex $d$-polytope with $f$-vector … where $f_k$ is the number of $k$-dimensional faces of $P$. The unimodality conjecture. For each $d$-polytope $P$ there is an inte…

Spanning conjecture. The set of $f$-vectors of all ordinary $d$-polytopes spans the Euler hyperplane. A spanning set consists of the ordinary polytopes

Let $\Delta$ be a $(d-1)$-dimensional pure flag complex on $2d$ vertices, and let its $f$-vector be the vector counting faces by dimension. Levit–Mandrescu conjecture. The $f$-…

Excess bound conjecture. If $v\notin\mathcal{E}^d$, then

Let $d$ be a positive integer. For a cubical $d$-polytope $Q$, let $g^c(Q)$ be its cubical $g$-vector, and let $\mathcal{C}_d$ be the minimal closed cone containing all such vector…

Let $\mathcal{F}(\mathcal{P}^4)$ denote the set of $f$-vectors of all $4$-polytopes, and let $\mathbin{{\scriptstyle\boxplus}”}$ be the modified addition realized by connected sum…

The f-vector comparison conjecture. One has $f_i\leq f'_i$ for all $1\leq i\leq d-1$. Moreover, if $f_i=f'_i$ for some $1\leq i\leq d-1$, then ${\mathcal O}(P)$ and…

Cook–Nagel and Constantinescu–Varbaro conjecture. The following equality of sets holds:

Let $K$ be a field, and consider the $h$-vectors of quadratic Artinian $K$-algebras and the $f$-vectors of simplicial complexes. Eisenbud–Green–Harris conjecture. Every $h$-vector…

A five-ball is a triangulated ball of dimension five, with $h$-vector written as $h=(1,h_1,h_2,h_3,h_4,h_5,0)$. Theorem's construction gives a collection of $h$-vectors described b…

Let $K$) be a connected triangulated $d$-manifold with $d\geq 3$, and let $g_2(K)$ denote its second $g$-number and $\beta_1(K;\mathbb{Q})$ its first Betti number over $\mathbb{Q}$…

A homology sphere is a pure simplicial complex $L$ such that every face link has the homology of a sphere of the same dimension. Its $g$-vector is the sequence derived from its $h$…

Let $L$ be a simplicial sphere, with $g$-vector defined from its $h$-vector by $g_0=h_0$ and $g_i=h_i-h_{i-1}$ for $i\geq 1$. An $M$-sequence is the Hilbert function of a standard…

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

Toric $g$-vector conjecture. The toric $g$-vector of a non-simplicial polytope is an $M$-sequence, i.e. it satisfies Macaulay inequalities.