  For a fixed $r$, let $f_r(n)$ denote the minimum number of complete
$r$-partite $r$-graphs needed to partition the complete $r$-graph on $n$
vertices. The Graham-Pollak theorem asserts that $f_2(n)=n-1$. An easy
construction shows that $f_r(n) \leq (1+o(1))\binom{n}{\lfloor r/2 \rfloor}$,
and we write $c_r$ for the least number such that $f_r(n) \leq c_r
(1+o(1))\binom{n}{\lfloor r/2 \rfloor}$.
  It was known that $c_r < 1$ for each even $r \geq 4$, but this was not known
for any odd value of $r$. In this short note, we prove that $c_{295}<1$. Our
method also shows that $c_r \rightarrow 0$, answering another open problem.
