  A multigraph $G$ is an $(s,q)$-graph if every $s$-set of vertices in $G$
supports at most $q$ edges of $G$, counting multiplicities. Mubayi and Terry
posed the problem of determining the maximum of the product of the
edge-multiplicities in an $(s,q)$-graph on $n$ vertices. We give an asymptotic
solution to this problem for the family $(s,q)=(2r,
a\binom{2r}{2}+\mathrm{ex}(2r, K_{r+1})-1 )$ with $r, a\in \mathbb{Z}_{\geq
2}$. This greatly generalises previous results on the problem due to Mubayi and
Terry and to Day, Treglown and the author, who between them had resolved the
special case $r=2$. Our result asymptotically confirms an infinite family of
cases in (and overcomes a major obstacle to a resolution of) a conjecture of
Day, Treglown and the author.
