39 problems

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…

Flag-complex partial-link conjecture.

Finite flag subcomplex vanishing conjecture.

Flag complex conjecture.

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 $\Delta$ 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 $f$-…

Complete-bipartite spanning-subgraph conjecture. If $G\in\mathbf{G}(n,m)$ and

Total-persistence conjecture. The extremal filtrations described in Section 4 achieve the maximal total persistence over any edgewise filtration of flag complexes in homology degre…

Let $G_n$ be a complete digraph on $n$ nodes, and let $\dFl(G_n)$ denote its flag complex. Write $H_k(\dFl(G_n))$ for the $k$-th homology group, and let $!n$ be the number of deran…

Let $K$ be a flag simplicial sphere, and let $\gamma(K)$ denote its gamma vector. A simplicial complex is balanced if, for its vertex set $V$, there is a map $\kappa:V\to[d]$ such…

Let $\mathcal{K}$ be a flag simplicial complex. The spaces $\mathrm{DJ}(\mathcal{K})$ and $\mathcal{Z}_\mathcal{K}$ are defined by the Davis–Januszkiewicz and moment-angle construc…

Chudnovsky–Nevo conjecture.

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…

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…

Turán lower-bound conjecture. The face vector of $\Delta$ is componentwise at least that of $T$:

Canonical-representation lower-bound conjecture. For every $i\geq 0$,

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…

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 $m\geq 2$, and let $\Delta$ be a flag homology $(2m-1)$-manifold with $n$ vertices. The Gal's stability conjecture. There exists a constant $b=b(m)$ such that if … then…

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 $\Delta$ be a flag homology sphere. Write its $\gamma$-vector as $\gamma^\Delta=(1,\gamma_1,\gamma_2,\ldots)$. A simplicial complex is flag when its faces are exactly the cliqu…

Flag Eulerian upper bound conjecture. For every $1\leq i\leq 2m-1$,