14 problems

Let $K_v^{(3)}$ be the complete $3$-uniform hypergraph on vertex set $\mathbb{Z}_v$, and let $I$ consist of the missing triplets $(i,i+v/3,i+2v/3)$ whenever $3\mid v$. A decomposit…

Let $q>r\geq 3$ be integers. An $r$-graph is an $r$-uniform hypergraph, and it is $K_q^r$-divisible when it satisfies the divisibility conditions necessary for a decomposition into…

Let $n>q>r\geq 1$ be integers. An $(n,q,r)$-Steiner system is a collection of $q$-subsets of an $n$-element set such that every $r$-subset is contained in exactly one member. Equiv…

Glock–Kühn–Osthus conjecture. For each integer $r\ge 3$, there exists a constant $C>0$ such that, for all integers $q>r$ and all sufficiently large $n$, every $K_q^r$-divisible $r$…

Reflection-symmetric strengthening. There exists a set $\Pi=\{\pi_1,\pi_2,\ldots,\pi_{C_k}\}$ of $C_k$ permutations of $\mathbb{Z}_{2k+1}$, each fixing $0$, and a bijection…

Dyck-path strengthening. There exists a set $\Pi=\{\pi_1,\pi_2,\ldots,\pi_{C_k}\}$ of $C_k$ permutations of $\mathbb{Z}_{2k+1}$, each fixing $0$, and a bijection…

The wreath conjecture. For any positive integers $k\leq n$, there is a decomposition of $\mathbb{Z}_{n}^{(k)}$ into disjoint wreaths.

A Steiner system with parameters $(n,q,r)$ is a design in which every $r$-subset belongs to exactly one $q$-subset. For a partial $(n,q,r)$-Steiner system, the girth is the smalles…

Let $k,n\in\mathbb{N}\setminus\{1\}$. Let $\mathcal{T}$ be a family of $k$-trees, where a $k$-tree is the recursively defined $k$-uniform tree obtained from one edge by repeatedly…

Let $k,n\in\mathbb{N}\setminus\{1\}$. Let $T$ be a $k$-tree, meaning a $k$-uniform tree defined recursively by starting with one edge and successively adding a vertex together with…

Let $k,n\in\mathbb{N}\setminus\{1\}$. A $k$-tree is defined recursively: a single edge is a $k$-tree, and a $k$-tree with $\ell$ edges is obtained from one with $\ell-1$ edges by a…

Minimum-degree Baranyai conjecture. For all sufficiently large $n$, an $n$-vertex $k$-graph $G$ with

Threshold equality conjecture. For all $r>k\ge 2$,

Vanishing-$\alpha_r$ conjecture. We have