Balanced partite decompositions of dense uniform hypergraphs

Determine whether decomposition results analogous to the graph biclique decompositions can be obtained for dense d-uniform hypergraphs using more balanced d-cliques, specifically d-cliques whose every part has size substantially larger than one, rather than decompositions in which all but two parts are singletons.

Background

The paper develops asymptotically optimal decompositions of d-uniform hypergraphs into complete d-partite hypergraphs, but its construction produces highly unbalanced d-cliques: all but two parts have size one. The authors note that the hypergraph Kővári–Sós–Turán theorem guarantees d-cliques whose parts each have size Ω((lg n){1/(d-1)}) in dense d-uniform hypergraphs, while existing algorithmic results address the task of finding such structures. The unresolved issue is whether decomposition methods with similarly balanced parts can achieve bounds comparable to those established for the highly unbalanced decompositions.

References

An interesting line of research is whether our results for hypergraphs can help find large balanced $d$-cliques in dense $d$-uniform hypergraphs. Concretely, $d$-cliques with each part having size $Omega( (lg n){1/(d-1)})$ are ensured by (the hypergraph version of KST), and the best algorithmic results are given by a recent result of Espuña . In contrast, our hypergraph decomposition results (\Cref{theorem:ces-hypergraph,theorem:ep-hypergraph-upper}) yield very unbalanced $d$-cliques, in which all but $2$ parts have size $1$. Thus, an interesting question is whether similar decomposition results can be obtained with more balanced $d$-cliques.

— Optimal and Efficient Partite Decompositions of Hypergraphs  (2511.11855 - Krapivin et al., 14 Nov 2025) in Section 6, Further directions, second bullet