On stability of the Erdős–Rademacher problem

Abstract

Mantel’s theorem states that every n-vertex graph with $\lfloor \frac{n^2}{4}\rfloor +t$ 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 $t\lfloor n∕2\rfloor$ 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.