Baranyai–Katona wreath decomposition conjecture

Decompose the complete family of all k-element subsets of an n-element cyclic set into pairwise disjoint wreaths for every pair of positive integers k and n with k ≤ n, where a wreath is generated by a permutation of the cyclic set as defined in the paper.

Background

The paper defines a wreath generated by a permutation of the cyclic group Z_n as the collection of k-element blocks obtained from consecutive length-k segments of the permutation. Each wreath has size n divided by gcd(n,k).

The original conjecture, attributed to Baranyai and Katona, asks whether the complete family of k-subsets can always be partitioned into such wreaths. The authors identify this conjecture, together with two stronger variants for n = 2k + 1, as the main open questions of the paper.