@article{Razborov2008OnTM,
title={On the Minimal Density of Triangles in Graphs},
author={Alexander A. Razborov},
journal={Combinatorics, Probability & Computing},
year={2008},
volume={17},
pages={603-618}
}

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… CONTINUE READING