A note on extremal constructions for the Erdős-Rademacher problem

For given positive integers $r\ge 3$ , $n$ and $e\le \binom{n}{2}$ , the famous Erd & odblac;s-Rademacher problem asks for the minimum number of $r$ -cliques in a graph with $n$ vertices and $e$ edges. A conjecture of Lov & aacute;sz and Simonovits from the 1970s states that, for every $r\ge 3$ , if $n$ is sufficiently large then, for every $e\le \binom{n}{2}$ , 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.