Explicit Baranyai Partitions for Quadruples, Part I: Quadrupling Constructions

TL;DR

This paper introduces an explicit recursive construction for Baranyai partitions when k=4 and n=4t, providing a linear-time method for certain values of t, addressing the non-explicit nature of previous existence proofs.

Contribution

It presents the first explicit recursive quadrupling construction for Baranyai partitions at k=4 for specific values of n, enabling efficient generation.

Findings

Provides a linear-time recursive construction for k=4, n=4t, with t in specific residue classes.

Addresses the non-explicit nature of Baranyai's theorem for k=4.

Sets the stage for future work on remaining cases in follow-up paper.

Abstract

It is well known that, whenever $k$ divides $n$, the complete $k$-uniform hypergraph on $n$ vertices can be partitioned into disjoint perfect matchings. Equivalently, the set of $k$-subsets of an $n$-set can be partitioned into parallel classes so that each parallel class is a partition of the $n$-set. This result is known as Baranyai's theorem, which guarantees the existence of \emph{Baranyai partitions}. Unfortunately, the proof of Baranyai's theorem uses network flow arguments, making this result non-explicit. 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 $k=2$ and $k=3$. In this paper, we present an explicit recursive quadrupling construction for $k=4$ and $n=4t$, where $t \equiv 0,3,4,6,8,9 ~(\text{mod}~12)$. In a follow-up paper (Part II), the other values of~$t$, namely $t \equiv 1,2,5,7,10,11 ~(\text{mod}~12)$, will be considered.