On stability of the Erd's-Rademacher problem

Mantel's theorem states that every n-vertex graph with [n(2)/4 ] + t edges, where t > 0, contains a triangle. The problem of determining the minimum number of trian-gles in such a graph is usually referred to as the Erdos-Rademacher problem. Lovasz 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 conjec-ture, 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.