Baranyai's theorem

In combinatorial mathematics, Baranyai's theorem (proved by and named after Zsolt Baranyai) deals with the decompositions of complete hypergraphs.

The statement of the result is that if ${\displaystyle 2\leq r<k}$ are natural numbers and r divides k, then the complete hypergraph ${\displaystyle K_{r}^{k}}$ decomposes into 1-factors. ${\displaystyle K_{r}^{k}}$ is a hypergraph with k vertices, in which every subset of r vertices forms a hyperedge; a 1-factor of this hypergraph is a set of hyperedges that touches each vertex exactly once, or equivalently a partition of the vertices into subsets of size r. Thus, the theorem states that the k vertices of the hypergraph may be partitioned into subsets of r vertices in ${\displaystyle {\binom {k}{r}}{\frac {r}{k}}={\binom {k-1}{r-1}}}$ different ways, in such a way that each r-element subset appears in exactly one of the partitions.

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