15 problems

Let $\Delta$ be a flag simplicial sphere, and let $\Gamma$ be a simplicial complex whose $f$-vector is $\gamma(\Delta)$. Nevo–Petersen balanced conjecture. The complex $\Gamma$ can…

Let $P$ be a balanced simplicial $d$-polytope, meaning that its underlying graph is $d$-colorable. For $-1\leq i\leq d-1$, let $f_i(P)$ be the number of $i$-dimensional faces, with…

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 $\Delta$ 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 $H_{\mathrm{bal}}$ be the cone generated by the $h$-vectors of balanced $d$-polytopes or balanced homology $(d-1)$-spheres, and let $H_{\mathrm{flag}}$ be the cone generated by…

Let $\Delta$ be a balanced $d$-polytope, or more generally a balanced $\mathbf{k}$-homology $(d-1)$-sphere. Balanced GLBC. … The inequalities were established for all balanced poly…

Let $\Delta$ be a balanced normal pseudomanifold of dimension $d-1\geq 3$. Klee–Novik's balanced lower bound conjecture. … This extends the balanced Lower Bound Theorem by includin…

Let $\Delta$ be a balanced, orientable $\mathbf{k}$-homology manifold without boundary. Define $g”_j(\Delta)=h”_j(\Delta)-h”_{j-1}(\Delta)$. Balanced manifold $g”$-conjecture. The…

Let $\Delta$ be a balanced connected ${\bf k}$-homology manifold that is orientable over ${\bf k}$ of dimension $d-1 \geq 3$. Let $\mathcal{BH}^d$ denote the class of balanced conn…

Let $\Delta$ be a balanced connected simplicial complex of dimension $d-1 \geq 3$. If $\Delta$ is a ${\bf k}$-homology manifold that is orientable over ${\bf k}$, then the balanced…

Let $K$ be a $d$-dimensional balanced simplicial complex, and let $<$ be a $(3,\ldots,3)$-admissible order. Let $K^{b,<}$ denote the corresponding balanced shift. Balanced linkless…

Let $K$ be a $d$-dimensional balanced complex, and let $\mathfrak{o}(K)$ denote its van Kampen obstruction to PL embeddability in $\mathbb{S}^{2d}$. Balanced van Kampen-obstruction…

Let $K$ be a $d$-dimensional balanced complex, and let $<$ be a $(2,\ldots,2)$-admissible order. Let $K^{b,<}$ denote the corresponding balanced shift. Balanced PL-embeddability co…

Let $K$ be a $d$-dimensional balanced complex, and let $<$ be a $(2,2,\ldots,2)$-admissible order, meaning that the least two vertices of each color class form an initial segment o…

Let $K$ be a $d$-dimensional balanced simplicial complex, meaning that its vertices are colored with $d+1$ colors so that every edge has vertices of different colors. Let $K^b$ den…