HOW MANY TRIANGLES CAN A GRAPH HAVE?

We investigate the set T-n of the possible number of triangles in a graph on n vertices. The first main result says that every natural number less than ((n)(3))-(root 2+o(1))n(3/2) belongs to T-n. On the other hand, we show that there is a number m=((n)(3))-(root 2+o(1))n(3/2) that is not a member of T-n. In addition, we prove that there are two interlacing sequences ((n)(3))-(root 2+o(1))n(3/2)=c(1)<= d(1)<= c(2)<= d(2)<=& ctdot;<= c(s)<= d(s)=((n)(3)) with divided by c(t)-d(t)divided by=n-2-((s-t+1)(2)) such that (c(t),d(t))boolean AND T-n=& empty; for all t. Moreover, we obtain a generalisation of these results for the set of the possible number of copies of a connected graph H in a graph on n vertices.