Baranyai's theorem

Baranyai's theorem states that if a positive integer r divides n, the complete r-uniform hypergraph on n vertices can be decomposed into 1-factors, each partitioning the vertices into r-element sets.

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Uniform hypergraph: A hypergraph whose edges all contain the same number of vertices. The theorem concerns the complete uniform hypergraph whose edges are all r-element subsets.
Perfect matching: A matching that covers every vertex of a graph. A 1-factor is the hypergraph counterpart of a perfect matching.
Social golfer problem: A scheduling problem that arranges golfers into groups while limiting repeated pairings. Baranyai factors provide rounds of equal-sized groups covering everyone exactly once.
Zsolt Baranyai: A Hungarian mathematician known for results in combinatorics, including the hypergraph factorization theorem bearing his name. Baranyai proved the theorem that establishes the asserted decomposition.
Complete hypergraph: A hypergraph containing every possible subset of a fixed size as an edge. Baranyai's theorem partitions the edges of this maximally populated hypergraph.
Divisibility: An integer relation in which one number is an exact multiple of another. The condition r divides n is necessary for each factor to cover all vertices.
Round-robin tournament: A competition format in which every participant or team meets every other one. Match schedules can be organized using factorizations of complete graphs and related designs.
Extremal combinatorics: The study of the largest or smallest structures satisfying specified combinatorial conditions. The theorem belongs to a field concerned with existence and structure of combinatorial configurations.
Hypergraph matching: A set of hyperedges in which no two edges share a vertex. Each 1-factor in the decomposition is a matching that covers every vertex.
Factorial: The product of the positive integers up to a given integer. Factorials appear when counting partitions into equally sized, unlabeled blocks.