The number of complete subgraphs of equi-partite graphs Original Research Article

Denote by Gr(n) = Gr(n,n,…,n) an r-partite graph having n vertices in each of the classes V1, V2, …, Vr. How large should the minimal value δr(n) be to guarantee that Gr(n) contains a Kr, a complete graph of order r, when δ(Gr(n)) > δr(n)? This problem has been studied by several authors, including Bollobás, Erdős and Straus (1974), Bollobás, Erdős and Szemerédi (1974), Graver (see Ballobás, Erdős and Straus (1974) and Jin (1992). But what is the maximal integer fr(n) such that Gr(n) contains at least fr(n) copies of Kr when δ(Gr(n)) = δr(n) + 1? It is trivial to see f2(n) = n. Bollobás, Erdős and Szemerédi (1974) showed that f3(n) = min(4, n). In this paper we shall show that f(in4)(n) = Θ(n3).