The maximum number of triangles in a K4-free graph

Let v, e and t denote the number of vertices, edges and triangles, respectively, of a K-4-free graph. Fisher (1988) proved that t less than or equal to(e/3)(3/2), independently of v. His bound is attained when e=3k(2) for some integer k, but not in general. We find here, for any given value of e, the maximum possible value of t. Again, the maximum does not depend on t,. We also show that certain 'gaps' occur in the set of triples (v,e,t) realizable by K-4-free graphs. (C) 1999 Elsevier Science B.V. All rights reserved.