21 problems

For each $n$, consider the set of unlabelled isomorphism classes of trees with $n$ nodes. For a tree $T$, let $\mathcal{B}(T)$ be its graphical building set and let…

Nonnegativity conjecture for discrete pseudomanifolds.

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 flag simplicial sphere of dimension $d-1$, and let … be the coefficients in its gamma expansion. Nevo–Petersen conjecture. There exists a simplicial complex…

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…

Lutz–Nevo conjecture. The following are equivalent:

Gamma-zero characterization conjecture. If $k\in\mathbb{N}$ and $G$ is a $k$-connected graph on $n$ vertices, then

Ohsugi–Tsuchiya conjecture. For every $i\geq 0$,

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 $d\geq 4$ be an integer and let $\Delta$ be a flag simplicial $(d-1)$-sphere. Nevo–Lutz's gamma-two conjecture. The following are equivalent: 1. $\gamma_2(\Delta)=0$. 2. There…

Let $\Delta$ be a flag homology sphere, and let $\gamma(\Delta)$ denote its $\gamma$-vector. Nevo's strengthening of Gal's conjecture. The vector $\gamma(\Delta)$ is the $f$-vector…

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…

Let $d=2k\geq 4$, and let $\Delta$ be a flag homology $(d-1)$-sphere on $n$ vertices. Let $\mathcal{J}_k(n)$ be the flag sphere obtained by joining $k$ graph cycles, each having ei…

Let $\Delta$ and $\Delta'$ be flag homology spheres, and suppose that $\Delta'$ geometrically subdivides $\Delta$. Let $\gamma(\Delta)$ and $\gamma(\Delta')$ denote their gamma-vec…

Let $V$ be a $d$-element set, and let $\Gamma$ be a flag vertex-induced homology subdivision of the simplex $2^V$. The local $h$-polynomial has a unique expansion … and…

Athanasia's conjecture. One has

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 $\Delta$ be a flag homology sphere, and let $\Delta'$ be a flag vertex-induced homology subdivision of $\Delta$. Let $\gamma(\Delta)$ and $\gamma(\Delta')$ denote their…

Let $\Delta$ and $\Delta'$ be flag homology spheres, and suppose that $\Delta'$ geometrically subdivides $\Delta$. Let $\gamma(\Delta)$ and $\gamma(\Delta')$ denote their $\gamma$-…