9 problems

Let $k\geq 2$. A flag sphere is a simplicial sphere with no missing faces of dimension larger than one, and $g_0,g_1,\ldots,g_k$ are its $g$-numbers. Charney–Davis conjecture. Ever…

Lutz–Nevo conjecture. The following are equivalent:

Let $\mathcal K$ be a flag $(d-1)$-sphere, let $G(\mathcal K)$ be its graph, let $f_0(\mathcal K)$ denote its number of vertices, and let $\alpha(G(\mathcal K))$ denote the size of…

Let a flag sphere be a flag $(d-1)$-sphere with $n$ vertices, and let $\alpha(\Delta)$ denote the maximum size of an independent set in its graph. Independence-number conjecture. F…

Let $\Delta$ be a flag simplicial sphere, and let $\Gamma$ be an induced equator of $\Delta$, meaning an induced subcomplex that is a codimension-$1$ sphere and is not the link of…

Let $J_m^*(n)$ be the suspension of a join of $m$ balanced cycles as defined in the source, and let $\mathcal{J}_m^*(n)$ be the family of joins $S*C_2*\cdots*C_m$ with $S$ a flag 2…

Let $J_m(n)$ be the simplicial $(2m-1)$-sphere on $n$ vertices obtained as the join of $m$ cycles, each having either $\lfloor n/m\rfloor$ or $\lceil n/m\rceil$ vertices. Flag uppe…

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 $(d-1)$-sphere, and let $g_i(\Delta)$ denote its $g$-numbers. Flag lower bound conjecture. … Equality holds if and only if $\Delta$ is the boundary…