Explicit Baranyai partitions for quadruples, Part I: Quadrupling constructions

Abstract

AbstractIt is well known that, whenever divides , the complete ‐uniform hypergraph on vertices can be partitioned into disjoint perfect matchings. Equivalently, the set of ‐subsets of n‐set can be partitioned into parallel classes so that each parallel class is a partition of the ‐set. This result is known as Baranyai's theorem, which guarantees the existence of Baranyai partitions. Unfortunately, the proof of Baranyai's theorem uses network flow arguments, making this result nonexplicit. In particular, there is no known method to produce Baranyai partitions in time and space that scale linearly with the number of hyperedges in the hypergraph. It is desirable for certain applications to have an explicit construction that generates Baranyai partitions in linear time. Such an efficient construction is known for and 3. In this paper, we present an explicit recursive quadrupling construction for and , where . In a follow‐up paper (Part II), the other values of , namely, , will be considered.