12 problems

Kalai's conjecture. If

Zheng's conjecture. For every $k$-uniform hypertree $T$ with $k\geq 3$, the multiplicity of the zero Laplacian eigenvalue is $k^{|E(T)|(k-2)}$. This conjecture concerns the Laplaci…

A hypertree is a $3$-uniform hypergraph in which every two vertices are connected by a unique path. A Steiner triple system is a $3$-uniform hypergraph in which every pair of verti…

The $NC(m,d)$ extremal conjecture. For $G\in NC(m,d)$,

Let ${\mathcal T_{n,d}}$ denote the family of $n$-vertex $d$-hypertrees, and let ${\mathcal C_{n,d}}$ denote the set of $n$-vertex $d$-collapsible $d$-hypertrees. Here, a $d$-colla…

Let $H=(V,E)$ be a hypertree, meaning a connected hypergraph without cycles. For an integer $k\ge 2$, a vertex labeling $f:V\to\mathbb{Z}_k$ induces an edge labeling by…

Let $H=(V,E)$ be a hypertree, meaning a connected hypergraph without cycles. A vertex labeling $f:V\to\mathbb{Z}_2$ induces an edge labeling by $f(e)=\sum_{v\in e}f(v)$ modulo $2$.…

Let $H=(V,E)$ be a hypertree, and let $X_H$ denote its chromatic symmetric function. The $F$-basis is the fundamental quasisymmetric-function basis. Hypertree F-positivity conjectu…

Let $n$ be the number of vertices, and let a 3-uniform edge-maximal hypertree be a 3-uniform hypertree to which no further edge can be added without destroying the hypertree proper…

Let $k$ be a positive integer, and let $\mathcal{F}=(V,\mathcal{E})$ be a $k$-uniform edge-minimal hypertree on $n=|V|$ vertices. Edge-minimal hypertree upper-bound conjecture. One…

Let $\overline{\mathcal{M}}_{0,n}$ be the moduli space of stable $n$-pointed genus-zero curves, and let $D_\Gamma$ be the divisor associated with an irreducible hypertree $\Gamma$…

Generalized Hovey conjecture. All $p$-uniform hypertrees are $k$-cordial for all $k$.