Let frβ(n) represent the minimum number of complete r-partite r-graphs
required to partition the edge set of the complete r-uniform hypergraph on
n vertices. The Graham-Pollak theorem states that f2β(n)=nβ1. An upper
bound of (1+o(1))(β2rββnβ) was known. Recently
this was improved to 1514β(1+o(1))(β2rββnβ) for even rβ₯4. A bound of
[2rβ(1514β)4rβ+o(1)](1+o(1))(β2rββnβ) was also proved recently. The smallest odd r
for which crβ<1 that was known was for r=295. In this note we improve
this to c113β<1 and also give better upper bounds for frβ(n), for small
values of even r.Comment: 8 page