Decomposition of the completer-graph into completer-partiter-graphs

For n > r > 1, let f,(n) denote the minimum number q, such that it is possible to partition all edges of the complete r-graph on n vertices into q complete r-partite r-graphs. Graham and Pollak showed that fz(n) = n-1. Here we observe that f3(n) = n-2 and show that for every fixed r > 2, there are positive constants cx(r) and c2(r) such that q(r) < f,(n)" n-f'/2J < c2(r) for all n > r. This solves a problem of Aharoni and Linial. The proof uses some simple ideas of linear algebra.