  The supersaturation problem for a given graph $F$ asks for the minimum number
$h_F(n,q)$ of copies of $F$ in an $n$-vertex graph with $ex(n,F)+q$ edges.
Subsequent works by Rademacher, Erd\H{o}s, and Lov\'{a}sz and Simonovits
determine the optimal range of $q$ (which is linear in $n$) for cliques $F$
such that $h_F(n,q)$ equals the minimum number $t_F(n,q)$ of copies of $F$
obtained from a maximum $F$-free $n$-vertex graph by adding $q$ new edges. A
breakthrough result of Mubayi extends this line of research from cliques to
color-critical graphs $F$, and this was further strengthened by Pikhurko and
Yilma who established the equality $h_F(n,q)=t_F(n,q)$ for $1\leq q\leq
\epsilon_F n$ and sufficiently large $n$. In this paper, we present several
results on the supersaturation problem that extend beyond the existing
framework. Firstly, we explicitly construct infinitely many graphs $F$ with
restricted properties for which $h_F(n,q)<q\cdot t_F(n,1)$ holds when $n\gg
q\geq 4$, thus refuting a conjecture of Mubayi. Secondly, we extend the result
of Pikhurko-Yilma by showing the equality $h_F(n,q)=t_F(n,q)$ in the range
$1\leq q\leq \epsilon_F n$ for any member $F$ in a diverse and abundant graph
family (which includes color-critical graphs, disjoint unions of cliques $K_r$,
and the Petersen graph). Lastly, we prove the existence of a graph $F$ for any
positive integer $s$ such that $h_F(n,q)=t_F(n,q)$ holds when $1\leq q\leq
\epsilon_F n^{1-1/s}$, and $h_F(n,q)<t_F(n,q)$ when $n^{1-1/s}/\epsilon_F\leq
q\leq \epsilon_F n$, indicating that $q=\Theta(n^{1-1/s})$ serves as the
threshold for the equality $h_F(n,q)=t_F(n,q)$. We also discuss some additional
remarks and related open problems.
