58 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…

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…

Barcode cardinality conjecture. The number of intervals in the barcode satisfies

Let $V$ be a finite set, let $2^V$ denote the simplex on $V$, and let $\Gamma$ be a flag triangulation of $2^V$. Let $\ell_V(\Gamma,x)$ denote its local $h$-polynomial. Athanasiadi…

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 doubly Cohen–Macaulay flag simplicial complex of dimension $d-1$, and let $h_\Delta=(h_0,\ldots,h_d)$ be its $h$-vector. Suppose that the equality case of the coe…

Let $K$ be a flag complex, and let $r$ be the number of vertices in a largest face of $K$. The face vector of $K$ is the vector of face numbers of $K$. Kalai–Eckhoff conjecture. If…

Let $L$ be a flag complex of dimension $n-1=2k-1$, and let $W_L(t)$ be its growth function. Define … where $h_L(t)$ is the $h$-polynomial of $L$. The flag complex conjecture. One s…

Flag-complex partial-link conjecture.

Finite flag subcomplex vanishing conjecture.

Flag complex conjecture.

Let $\mathcal{K}$ be a flag complex, let $\mathbf{a},\mathbf{b}\in\mathbb{Z}_{\geq 0}^{\mathcal{V}}$ have disjoint supports, and write $|\mathbf{b}|$ for the sum of the entries of…

Let $\Delta$ be a flag $4$-sphere with $n$ vertices. Its number of edges is $f_1(\Delta)$. Adam–Hladký's conjecture. Every flag $4$-sphere with $n$ vertices satisfies … This would…

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 $f$-…

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 $A$ be a vertex decomposable flag simplicial complex, and let $h(A)$ denote its $h$-vector. A face-vector conjecture. The vector $h(A)$ should be the face vector of some flag s…

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 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…

The gamma2 conjecture.