  In 1973 Bermond, Germa, Heydemann and Sotteau conjectured that if $n$ divides
$\binom{n}{k}$, 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 $v_1,e_1,v_2,\dots,v_n,e_n$ of distinct vertices $v_i$ and
distinct edges $e_i$ so that each $e_i$ contains $v_i$ and $v_{i+1}$. So the
divisibility condition is clearly necessary. In this note, we prove that the
conjecture holds whenever $k \ge 4$ and $n \ge 30$. Our argument is based on
the Kruskal-Katona theorem. The case when $k=3$ was already solved by Verrall,
building on results of Bermond.
