Lovász–Simonovits conjecture on the Erdős–Rademacher problem

Let $K_r$ denote the complete graph on $r$ vertices. For integers $n$ and $m$ with $0\leq m\leq {n\choose 2}$, let $G_r(n,m)$ be the minimum number of copies of $K_r$ in an $(n,m)$-graph, meaning a graph with $n$ vertices and $m$ edges. Let $\mathcal{H}$ consist of all graphs obtained from a complete multipartite graph by adding a triangle-free graph into one of its parts, and let $H_r(n,m)$ be the minimum number of $r$-cliques in an $(n,m)$-graph from $\mathcal{H}$. Lovász–Simonovits conjecture. For every integer $r\geq 3$, there is $n_0$ such that for all $n\geq n_0$ and $0\leq m\leq {n\choose 2}$, we have

$$G_r(n,m)\geq H_r(n,m).$$

Since every graph in $\mathcal{H}$ is an admissible $(n,m)$-graph, the reverse inequality $G_r(n,m)\leq H_r(n,m)$ is immediate; thus the conjecture asserts equality for sufficiently large $n$. It is a central open problem in the Erdős–Rademacher problem concerning the minimum number of cliques forced by a given number of edges.