Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) such that $V\neq \varnothing$ and $\varnothing \notin E$. A *matching* is a subset $M\subseteq E$ such that $m\_1\neq m\_2 \in M$ implies $m\_1\cap m\_2 = \varnothing$, and $M$ is said to be *perfect* if $\bigcup M = V$. We say that $H$ is *$1$-factorizable* if $E$ is the disjoint union of perfect matchings.


If $H$ is $1$-factorizable, it is easy to see that it implies that for any vertices $v\_1,v\_2\in V$ we have $\text{deg}(v\_1) = \text{deg}(v\_2)$ where for $v\in V$ we set $\text{deg}(v) = |\{e\in E: v\in e\}|$.


We regard every positive integer $n$ as an ordinal, that is $n = \{0, \ldots, n-1\}$. If $X$ is a set and $\kappa$ is a cardinal, we let $[X]^\kappa$ be the collection of subsets of $X$ having cardinality $\kappa$.


**Question.** If $k, n> 2$ are integers, is the hypergraph $(k\cdot n, [k\cdot n]^k)$ $1$-factorizable?


(The answer is yes for $k\leq 2$.)



$1$-factorizability for "complete" finite hypergraphs
This is [Baranyai's theorem](https://en.wikipedia.org/wiki/Baranyai%27s_theorem). Other than in Baranyai's original paper you can also find a cool proof in the article "Uniform hypergraphs" by Brouwer and Schrijver which uses max-flow min-cut.


