A non-uniform extension of Baranyai's Theorem

Abstract

A celebrated theorem of Baranyai states that when k divides n, the family K k n of all k-subsets of an n-element set can be partitioned into perfect matchings. In other words, K k n is 1-factorable. In this paper, we determine all n, k, such that the family K ≤k n 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 ≤k n is 1-factorable if and only if n ≡ 0 or −1 (mod k).