21 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 $2\leq i\leq d/2$, 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 $k\geq 2$, $d\geq 2k$, 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…

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.

Let $s(d,n)$ denote the number of simplicial $(d-1)$-spheres with $n$ labeled vertices, and let $\operatorname{sn}(d,n)$ denote the number of $\lfloor d/2\rfloor$-neighborly simpli…

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. A graph is $(d+1)$-rigid if it has the corresponding generic rigidity property. The $(d+1)$-rigidity…

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 $\Delta$ 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…

Let $\Sigma$ be a simplicial sphere of dimension $d-1$, with $f$-vector $(f_{-1},f_0,\ldots,f_{d-1})$ and associated $g$-vector defined by $g_k(\Sigma)=h_k-h_{k-1}$ for $k\leq d/2$…

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 $d\geq 4$ be even, and let $\Delta$ be a flag simplicial $(d-1)$-sphere on $n$ vertices. Let $f_{i-1}(r,m)$ denote the number of $i$-cliques in the complete $r$-partite Turán g…

Let $d\geq 4$ be an integer and let $\Delta$ be a flag simplicial $(d-1)$-sphere. Define $\gamma_2(\Delta)=f_1-(2d-3)f_0+2d(d-2)$. An edge contraction replaces an edge $e$ by the c…

Let $L$ be a homology $(d-1)$-sphere. Write $\binom{a}{b}$ only through the interval notation in the claim, where $[a,b]$ denotes the corresponding set of integers, and let … Delta…

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 $h$…