Decomposing the Complete $r$-Graph

Let $f_r(n)$ be the minimum number of complete $r$-partite $r$-graphs needed to partition the edge set of the complete $r$-uniform hypergraph on $n$ vertices. Graham and Pollak showed that $f_2(n)=n-1$. An easy construction shows that $f_r(n)\le (1-o(1))\left(\genfrac{}{}{0px}{}{n}{\lfloor r\mathrm{/}2\rfloor }\right)$ and it has been unknown if this upper bound is asymptotically sharp. In this paper we show that $f_r(n)\le (\frac{14}{15}+o(1))\left(\genfrac{}{}{0px}{}{n}{r\mathrm{/}2}\right)$ for each even $r\ge 4$.