On the decomposition of random hypergraphs

For an r-uniform hypergraph H, let f(H) be the minimum number of complete r-partite r-uniform subhypergraphs of H whose edge sets partition the edge set of H. For a graph G, f(G) is the bipartition number of G which was introduced by Graham and Pollak in 1971. In 1988, Erdős conjectured that if G∈G(n,1/2), then with high probability f(G)=n−α(G), where α(G) is the independence number of G. This conjecture and its related problems have received a lot of attention recently. In this paper, we study the value of f(H) for a typical r-uniform hypergraph H. More precisely, we prove that if (log⁡n)2.001/n≤p≤1/2 and H∈H(r)(n,p), then with high probability f(H)=(1−π(K(r−1)r)+o(1))(nr−1), where π(Kr(r−1)) is the Turán density of Kr(r−1).