A non-uniform extension of Baranyai's Theorem

TL;DR

This paper extends Baranyai's Theorem to non-uniform hypergraphs by characterizing when the family of all subsets up to size k can be partitioned into perfect matchings, revealing new conditions for 1-factorability.

Contribution

It determines all n, k for which the family of subsets up to size k is 1-factorable, extending Baranyai's Theorem to a non-uniform setting.

Findings

For fixed k and large n, K_n^{≤k} is 1-factorable iff n ≡ 0 or -1 mod k.

The paper characterizes 1-factorability for non-uniform families of subsets.

Provides a complete classification of when such families can be partitioned into perfect matchings.

Abstract

A celebrated theorem of Baranyai states that when $k$ divides $n$, the family $K_n^k$ of all $k$-subsets of an $n$-element set can be partitioned into perfect matchings. In other words, $K_n^k$ is $1$-factorable. In this paper, we determine all $n, k$, such that the family $K_n^{\le k}$ consisting of subsets of $[n]$ of size up to $k$ is $1$-factorable, and thus extend Baranyai's Theorem to the non-uniform setting. In particular, our result implies that for fixed $k$ and sufficiently large $n$, $K_n^{\le k}$ is $1$-factorable if and only if $n \equiv 0$ or $-1 \pmod k$.