  For given positive integers $r\ge 3$, $n$ and $e\le \binom{n}{2}$, the famous
Erd\H{o}s--Rademacher problem asks for the minimum number of $r$-cliques in a
graph with $n$ vertices and $e$ edges. A conjecture of Lov\'{a}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.
