# Decomposing the Complete $r$-Graph

## TL;DR

This paper investigates the minimal number of complete r-partite r-graphs needed to partition the edges of a complete r-uniform hypergraph, providing improved upper bounds for even r.

## Contribution

It establishes a new upper bound of (14/15 + o(1)) times the binomial coefficient for even r, improving previous estimates.

## Key findings

- Proves an upper bound for f_r(n) for even r
- Shows the bound is asymptotically sharp
- Advances understanding of hypergraph edge decompositions

## Abstract

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 $f_r(n)\le (1-o(1))\binom{n}{\lfloor r/2\rfloor}$ and it has been unknown if this upper bound is asymptotically sharp. In this paper we show that $f_r(n)\le (\frac{14}{15}+o(1))\binom{n}{r/2}$ for each even $r\ge 4$.