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

TL;DR

This paper provides an exact solution for the minimum number of triangles in large graphs with a given size and order, confirming a longstanding conjecture and describing the extremal graphs.

Contribution

It offers an exact solution for the Erd ext{"o}s-Rademacher problem for large graphs with bounded edge density, extending previous asymptotic results.

Findings

Confirmed a conjecture of Lovász and Simonovits from 1975.

Provided a description of extremal graphs for the problem.

Solved the problem exactly for all large graphs with edge density bounded away from 1.

Abstract

What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Tur\'an, Rademacher solved the first non-trivial case of this problem in 1941. The problem was revived by Erd\H{o}s in 1955; it is now known as the Erd\H{o}s-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 Lov\'asz and Simonovits from 1975. Furthermore, we give a description of the extremal graphs.