A note on extremal constructions for the Erdős--Rademacher problem

For given positive integers $r\ge 3$, $n$ and $e\le \left(\genfrac{}{}{0px}{}{n}{2}\right)$, the famous Erd\H os–Rademacher problem asks for the minimum number of $r$-cliques in a graph with $n$ vertices and $e$ edges. A conjecture of Lovász and Simonovits from the 1970s states that, for every $r\ge 3$, if $n$ is sufficiently large then, for every $e\le \left(\genfrac{}{}{0px}{}{n}{2}\right)$, at least one extremal graph can be obtained from a complete partite graph by adding a triangle-free graph into one part. In this note, we explicitly write the minimum number of $r$-cliques predicted by the above conjecture. Also, we describe what we believe to be the set of extremal graphs for any $r\ge 4$ and all large~$n$, amending the previous conjecture of Pikhurko and Razborov.