A non-uniform extension of Baranyai's Theorem

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 $-1(\mathrm{mod}k)$.