22 problems

A simplicial complex is balanced if its graph is properly $d$-colorable when it has dimension $d-1$. Kalai's flag Cohen–Macaulay conjecture. The $f$-vector of any flag Cohen–Macaul…

Let $\Delta$ be an orientable $(d-1)$-dimensional $\mathbf{k}$-homology manifold without boundary. Define the modified $h$-numbers $h_i”$ as in the source by … where…

Let $J_m(n)$ denote the join of $m$ suitably balanced cycles on $n$ vertices, the proposed extremal flag odd-dimensional sphere. Adamaszek–Hladký pseudomanifold upper bound conject…

Let $(h_0,\ldots,h_d)$ be a non-negative integer vector, and for each $k=0,1,\ldots,d+1$ let … with $h_i=0$ for $i>d$. Billera–Lee conjecture. If all these difference vectors are…

Higher-dimensional face lower bound conjecture. For $j=1,\dots,k$,

Let $\Delta$ be a homology sphere, and let $\mathbf{k}[\Delta]$ be its face ring over a field $\mathbf{k}$. The hard Lefschetz theorem concerns the existence of a suitable Artinian…

A simplicial polytope is a polytope whose boundary is a simplicial complex, and its face numbers record the number of faces in each dimension. McMullen's conjecture. The face numbe…

The algebraic $g$-conjecture. Every rational homology sphere has Lefschetz property.

The $g$-conjecture. The $g$-vector of a rational homology sphere is an $M$-vector.

Let $\Delta$ be a rational homology sphere. It has the strong Lefschetz property if there exist a linear system of parameters $\Theta$ for $\mathbb{Q}[\Delta]$ and a linear form…

Let $\Delta$ be a simplicial complex of dimension $d$. Assume that $\Delta$ admits a PL embedding into $\mathbb{R}^{2d}$. GrunbaumKalaiSarkaria conjecture. Then … This conjecture s…

Let $\Delta$ be a simplicial sphere or a rational homology sphere. Its $h$-vector is subject to the $g$-theorem conditions: Dehn–Sommerville symmetry, unimodality up to the middle,…

Let $J_m^*(n)$ be the suspension of a join of $m$ balanced cycles as defined in the source, and let $\mathcal{J}_m^*(n)$ be the family of joins $S*C_2*\cdots*C_m$ with $S$ a flag 2…

Let $J_m(n)$ be the simplicial $(2m-1)$-sphere on $n$ vertices obtained as the join of $m$ cycles, each having either $\lfloor n/m\rfloor$ or $\lceil n/m\rceil$ vertices. Flag uppe…

Let $h(\Delta)$ denote the $h$-vector of a simplicial complex and $f(\Delta)$ its $f$-vector. Flag Cohen–Macaulay h-vector conjecture. The set of $h$-vectors of flag Cohen–Macaulay…

Let $\Delta$ be a flag homology $(d-1)$-sphere, and let $g_i(\Delta)$ denote its $g$-numbers. Flag lower bound conjecture. … Equality holds if and only if $\Delta$ is the boundary…

Let $f(\text{--})$ denote the face vector of a simplicial complex, and let the $g$-theorem give the characterization of the $f$-vectors of simplicial polytopes. Spherical g-theorem…

Let $P$ be an $(r-1)$-stacked simplicial $d$-polytope, and let its stack subdivision be the subdivision obtained from the stacking construction. Murai–Nevo's shellability conjectur…

Let $(P,A,\mathbf{\lambda})$ be a marked poset, and let $(U_1,U_2)$ and $(V_1,V_2)$ be different admissible decompositions of $P\setminus A$, meaning that each pair partitions…

Let $(P,A,\mathbf{\lambda})$ be a marked poset. For an $N$-dimensional polytope $Q$, let $f_i(Q)$ denote the number of its $i$-dimensional faces, and write…

Let $(P,A,\mathbf{\lambda})$ be a regular marked poset, and let $\operatorname*{St}(P)$ denote the set of star elements. An admissible decomposition of $P\setminus A$ is a pair…

Let $\Delta$ be a simplicial (homology) sphere, with $h$-vector and $g$-vector as in the $g$-theorem. The $g$-conjecture for spheres. The conditions of the $g$-theorem should hold…