The minimum number of triangles in graphs of given order and size

© 2017 Elsevier B.V. In the 1940s and 50s, Erdős and Rademacher raised the quantitative question of determining the number of triangles one can guarantee in a graph of given order and size. This problem has garnered much attention and, in a major breakthrough, was solved asymptotically by Razborov in 2008, whose results were extended by Nikiforov and Reiher. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from one. This proves almost every case of a conjecture of Lovász and Simonovits from 1975.