34 problems

Let $\Delta$ be a flag simplicial $d$-sphere. Its second gamma-number $\gamma_2(\Delta)$ is defined by … where $f_0(\Delta)$ and $f_1(\Delta)$ are the numbers of vertices and edges…

Let $S(j,d-1)$ denote the class of $(\mathbb{Z}/2\mathbb{Z})$-homology $(d-1)$-spheres without missing faces of dimension larger than $j$. Nevo's conjecture. For $1\leq j\leq d$, s…

Let a 2-sphere be a triangulated sphere with $n$ vertices, not necessarily flag, and let $i(\Delta)$ denote its independent domination number, the minimum size of a maximal indepen…

Fix positive integers $n$ and $m$, and consider flag triangulations of the $(n-1)$-dimensional sphere with $m$ vertices. Let the $(n-2)$-fold suspension of the one-dimensional sphe…

Let $S_d(U)$ be a squeezed sphere, or an S-squeezed sphere, constructed from a shifted order ideal $U$. A simplicial sphere has the strong Lefschetz property when its Lefschetz map…

Kalai's conjecture. For every such shifted order ideal,

Let $2\leq i\leq d/2$, let $\Delta\in S(d-i,d-1)$, and let $p$ be a generic or natural embedding of $\Delta$ in $\mathbb{R}^d$. An affine $i$-stress is an element of the affine str…

Assume $u\leq d/2$. Let $(\Delta,p)$ be a $(d-1)$-sphere in $S(d-u,d-1)$, with $p$ a generic or natural embedding in $\mathbb{R}^d$. Let $\mathcal{S}^a_i(\Delta,p)$ denote the spac…

Let $S(2,4)$ be the class of $(\mathbb{Z}/2\mathbb{Z})$-homology $4$-spheres without missing faces of dimension larger than two, and let $g_1=f_0-6$. Nevo's lower-bound conjecture.…

Let $k\geq 2$, $d\geq 2k$, and let $\Delta$ be a $(d-1)$-sphere with $g_k(\Delta)=1$. Cell-complex realization conjecture. There exists a $d$-dimensional cell complex $\mathcal{C}$…

Let $d\geq 3$, let $\mathbb S^d$ be the $d$-dimensional sphere, and let $\operatorname{ex}_{\hom}(n,\mathbb S^d)$ denote its topological Turán number. Newman–Pavelka sphere exponen…

Let $\mathcal{K}$ be a three-dimensional simplicial sphere, and let $\mathcal{Z}_{\mathcal{K}}$ be its moment-angle manifold. Suppose that either $\mathcal{K}^1$ is a chordal graph…

Chudnovsky–Nevo conjecture.

Let $2\leq i\leq d/2$. Let $\Delta$ be either the boundary complex of a simplicial $d$-polytope with its natural embedding $p$, or a simplicial $(d-1)$-sphere with a generic embedd…

Affine partition-of-unity conjecture.

Affine partition-of-unity conjecture. For all $1\leq j<i\leq d/2$,

Let $G$ be the graph of a flag triangulation of the $(d-1)$-dimensional sphere on $n$ vertices, and let $\alpha_M(d,n)$ denote the maximum possible size of a stable set in such a g…

Let $G$ be the graph of a flag triangulation of the $(d-1)$-dimensional sphere on $n$ vertices, and let $I$ be a stable set of $G$. The half-density conjecture. For flag spheres, t…

Let $G$ be the graph of a flag triangulation of the $(d-1)$-dimensional sphere. A graph is $(d+1)$-rigid if it has the corresponding generic rigidity property. The $(d+1)$-rigidity…

Let $G$ be a graph whose clique complex is a flag triangulation of the $(d-1)$-dimensional sphere, and define … where $\alpha(G)$ is the size of a largest stable set of $G$. The ex…

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,…

The gamma2 conjecture.

Let $\mathcal{R}$ be the family of minimal flag homology spheres, excluding the octahedral spheres; equivalently, these are the flag homology spheres in which every edge belongs to…

Let $\Delta$ be a flag homology sphere. An equator is an induced subcomplex that is a flag homology sphere of codimension $1$. Equator Conjecture. For every equator $E$ in $\Delta$…

Let $\Delta$ be a flag homology sphere and let $v$ be a vertex. Its vertex link is $\operatorname{lk}_v(\Delta)$. Link Conjecture. The link's $\gamma$-vector is coefficientwise at…