On stability of the Erdős–Rademacher problem

Abstract

Mantel’s theorem states that every n-vertex graph with ⌊n24⌋+t edges, where t>0, contains a triangle. The problem of determining the minimum number of triangles in such a graph is usually referred to as the Erdős–Rademacher problem. Lovász and Simonovits proved that there are at least t⌊n∕2⌋ triangles in each of those graphs. Katona and Xiao considered the same problem under the additional condition that there are no s−1 vertices covering all triangles. They settled the case t=1 and s=2. Solving their conjecture, we determine the minimum number of triangles for every fixed pair of s and t, when n is sufficiently large. Additionally, solving another conjecture of Katona and Xiao, we extend the theory for considering cliques instead of triangles.