On the Minimal Density of Triangles in Graphs

The most famous result of extremal combinatorics is probably the celebrated theorem of Turán [20] determining the maximal number ex(n;Kr) of edges in a Kr-free graph on n vertices. Asymptotically, ex(n;Kr) ≈ ( 1− 1 r−1 ) ( n 2 ) . The non-trivial part (that is, the upper bound) of this theorem in the contrapositive form can be stated as follows: any graph G with m > ex(n;Kr) edges contains at least one copy of Kr. The quantitative version of this latter statement (that is, how many such copies, as a function fr(m,n) of r, n,m, must necessarily exist in any graph G) received quite a fair attention in combinatorial literature (in more general context of arbitrary forbidden subgraphs, questions of this sort were named in [7] “the theory of supersaturated graphs”) and turned out to be notoriously difficult. Erdös [4, 5] computed fr(m,n) exactly when m is very close to ex(n;Kr); more specifically, when m ≤ ex(n;Kr) + Crn (1.1)