16 problems

Let $\mathcal{T}$ be a tight $k$-tree on $v$ vertices, meaning a hypertree of $k$-sets in which consecutive edges intersect in $k-1$ elements, and $|\bigcup\mathcal{T}|=v$. Let…

Elliott–Rödl conjecture. Every hypertree of $(1-o(1))n$ vertices can be found in any Steiner triple system.

Generating-function conjecture. For each $k\geq 2$ and $\ell\geq 3$,

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…

Let $(T,\mathbf{w})$ be a weighted $k$-tree, and let $T'$ be a subtree of $T$. Write $\mu(T',\mathbf{w},x)$ for the weighted matching polynomial of $T'$ and let…

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

Let $d\geq 2$, and consider $d$-dimensional hypertrees with complete $(d-1)$-skeleton. A $d$-dimensional hypertree is a $d$-dimensional $\mathbb{Q}$-acyclic complex on a vertex set…

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…

A $k$-uniform hypertree is a hypergraph whose edges satisfy the hypertree conditions defined in the paper; an edge-minimal hypertree is one in which deleting any edge destroys the…

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

Non-acyclicity conjecture. $X_A$ is not $\mathbb{Z}$-acyclic for most $(k+1)$-subsets $A\subset \mathbb{Z}_n$.