Improved Bounds for the Graham-Pollak Problem for Hypergraphs

For a fixed r, let fr(n) denote the minimum number of complete r-partite rgraphs needed to partition the complete r-graph on n vertices. The Graham-Pollak theorem asserts that f2(n) = n − 1. An easy construction shows that fr(n) 6 (1 + o(1)) ( n br/2c ) , and we write cr for the least number such that fr(n) 6 cr(1 + o(1)) ( n br/2c ) . It was known that cr < 1 for each even r > 4, but this was not known for any odd value of r. In this short note, we prove that c295 < 1. Our method also shows that cr → 0, answering another open problem.