Decompositions of complete uniform hypergraphs into Hamilton Berge cycles

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In 1973 Bermond, Germa, Heydemann and Sotteau conjectured that if n divides nk, then the complete k-uniform hypergraph on n vertices has a decomposition into Hamilton Berge cycles. Here a Berge cycle consists of an alternating sequence v₁,e₁,v₂,,vₙ,eₙ of distinct vertices vᵢ and distinct edges eᵢ so that each eᵢ contains vᵢ and vᵢ₊₁. So the divisibility condition is clearly necessary. In this note, we prove that the conjecture holds whenever k ≥ 4 and n ≥ 30. Our argument is based on the Kruskal-Katona theorem. The case when k=3 was already solved by Verrall, building on results of Bermond.