Decompositions of complete uniform hypergraphs into Hamilton Berge cycles

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 _{e i} so that each _{e i} 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.