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

What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turán, the first non-trivial case of this problem was solved by Rademacher in 1941, and the problem was revived by Erdős in 1955; it is now known as the Erdős-Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from 1, which in this range confirms a conjecture of Lovász and Simonovits from 1975. Furthermore, we give a description of the extremal graphs