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Baranyai's theorem : ウィキペディア英語版
Baranyai's theorem

In combinatorial mathematics, Baranyai's theorem (proved by and named after Zsolt Baranyai) deals with the decompositions of complete hypergraphs.
==Statement of the theorem==
The statement of the result is that if 2\le r are natural numbers and ''r'' divides ''k'', then the complete hypergraph K^k_r decomposes into 1-factors. K^k_r 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 \binom\frac different ways, in such a way that each ''r''-element subset appears in exactly one of the partitions.

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