29 problems

Let $q>r\geq 2$ and let a girth-$g$ decomposition mean a $K_q^r$-decomposition containing no configuration of the forbidden size below $g$. High Girth Existence Conjecture. For eve…

Let $K_q^r$ be the complete $r$-uniform hypergraph on $q$ vertices, and let $\delta_{K_q^r}$ denote its decomposition threshold. Glock–Kühn–Osthus conjecture. For fixed $r$, the th…

Let $K_n^r$ be the complete $r$-uniform hypergraph and $K_q^r$ its complete $q$-vertex subhypergraph. A packing is a collection of edge-disjoint copies of $K_q^r$ in $K_n^r$. Erdős…

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

Let $3\leq k<n$, and let $K_n^{(k)}$ be the complete $k$-uniform hypergraph on $n$ vertices, with $\binom nk$ edges. A Hamilton Berge cycle is a Berge cycle of length $n$ using all…

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…

Let $n>q>r\geq 1$ and let $X$ be an $n$-set. A design with parameters $(n,q,r,\lambda)$ is a set of $q$-subsets of $X$ in which every $r$-subset of $X$ belongs to exactly $\lambda$…

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…

For positive integers $k\leq n$, let $\mathbb{Z}_{n}$ be the integers modulo $n$, and let $\mathbb{Z}_{n}^{(k)}$ denote the set of $k$-element subsets of $\mathbb{Z}_{n}$. For a pe…

Let $n_0\in\mathbb{N}$ and $C\in\mathbb{R}$ be constants. For every $n\geq n_0$ satisfying $n\equiv 1,3\pmod{6}$, let $\mathcal{H}=\{H_1,\dots,H_{n(n-1)/6}\}$ be a family of $3$-un…

A complete $r$-uniform hypergraph is a hypergraph in which every $r$-element subset of its vertices is an edge. A decomposition into copies of another hypergraph partitions the edg…

Let $H$ be a $3$-uniform hypergraph, let $|H|$ denote its number of vertices, and let $[?]$ be its minimum codegree. A cycle decomposition is a decomposition of the edges of $H$ in…

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…

Let $k\geq 3$, and let $n$ be sufficiently large in terms of $k$ with $k\mid\binom{n-1}{k-1}$. A tight cycle in a $k$-graph is a cycle whose vertices can be cyclically ordered so t…

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

Wreath decomposition conjecture. The complete hypergraph $K_n^k$ can always be decomposed into wreaths.

Bailey–Stevens conjecture. The complete hypergraph $K_n^k$ has a decomposition into tight Hamilton cycles if and only if

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

Existence conjecture. For fixed $r$ and $k$, and sufficiently large $n$, these divisibility conditions are sufficient for a $K_r^k$-decomposition of $K_n^k$.

Bailey–Stevens conjecture. The complete hypergraph $K_n^k$ has a decomposition into tight Hamilton cycles if and only if $n\mid\binom{n}{k}$.