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Existence and construction of k-factorizations of the complete k-uniform hypergraph when k does not divide n

Establish whether k-factorizations of the complete k-uniform hypergraph K_n^{(k)} exist for general n when k does not divide n, and, if they exist, develop efficient algorithms to construct such factorizations (equivalently, full k-equireplicate partitions of the block family B_k).

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Background

For k>2, building r-equireplicate designs relates to hypergraph factorization problems. When k divides n, Baranyai’s theorem guarantees the existence of certain factorizations, but general, sequential constructions remain difficult.

When k does not divide n, the task reduces to constructing a k-factorization of the complete k-uniform hypergraph K_n{(k)}. The existence, and especially efficient construction, of such factorizations is a longstanding open problem linked to cyclic decompositions and other combinatorial structures.

References

In this case, the problem is equivalent to constructing a k-factorization of the complete k-uniform hypergraph K_n{(k)}. However, even the existence of such a factorization is not guaranteed—let alone an efficient algorithm for its construction—as this remains an open problem in combinatorics (see e.g., that discusses cyclic decompositions of $K_n{(k)}$).

— Incomplete U-Statistics of Equireplicate Designs: Berry-Esseen Bound and Efficient Construction (2510.20755 - Miglioli et al., 23 Oct 2025) in Supplementary Section S4, subsection “Construction of r-equireplicate designs when k > 2”