Paths in graphs

We prove that if 10 less than or equal to ((k)(2)) less than or equal to m < ((k+1)(2)) then the number of paths of length three in a graph G of size m is at most 2m(m - k)(k - 2)/k. Equality is attained iff G is the union of KI, and isolated vertices. We also give asymptotically best possible bound s for the maximal number of paths of length s, for arbitrary s, in graphs o f size m. Lastly, we discuss the more general problem of maximizing the num ber of subgraphs isomorphic to a given graph H in graphs of size m.