Baranyai’s wreath conjecture

Establish whether the family of all k-element subsets of the cyclic group Z_n can be decomposed into pairwise disjoint wreaths from \(\mathcal{W}_{n,k}\) for every pair of positive integers \(k\leq n\).

Background

A wreath generated by a permutation of $\mathbb{Z}_n$ is a family of k-element subsets formed from consecutive blocks of length k in the permutation, with repetitions removed according to the cyclic construction. Baranyai’s conjecture asks whether all k-element subsets of $\mathbb{Z}_n$ admit a partition into such wreaths.

The paper reformulates this decomposition problem using the wreath matrix $M(n,k)$, proving that the conjecture is equivalent to the existence of a kernel vector satisfying specified sign and support conditions. The authors state that the conjecture remains unresolved in general, including for $k=3$, where it is known only for certain families of n.