On Subgraphs of the Complete Bipartite Graph

G(n) denotes a graph of n vertices and G(n) denotes its complementary graph. In a complete graph every two distinct vertices are joined by an edge. Let C k (G(n)) denote the number of complete subgraphs of k vertices contained in G(n). Recently it was proved [1] that for every k 2 (n) (1) min C (G (n)) + Ck(G(n)) < k k, , ! 2 2 where the minimum is over all graphs G(n). It seems likely that (1) is not far from being best possible and that (2) lim min n-0 C k (G(n)) + C k (G(n)) 2 Ik) ~21 2 That this is true for k = 3 follows from the results of Goodman [2], Sauvé [5], and Lorden [3]. We are unable to prove (2) for k > 3 but we can prove an analogous result for bipartite graphs. The bipartite graph B(m,n) consists of the vertices x1 ,. .. , xm and y 1 ,. .. y n and some of the edges (x y,). B(m,n) has the same vertices, and the edge (x,) y,) is in ~ J B(m, n) if it is not in B(m, n). If B(m, n) contains mr_