  The classic extremal problem is that of computing the maximum number of edges
in an $F$-free graph. In the case where $F=K_{r+1}$, the extremal number was
determined by Tur\'an. Later results, known as supersaturation theorems, proved
that in a graph containing more edges than the extremal number, there must also
be many copies of $K_{r+1}$. Alon and Shikhelman introduced a broader class of
problems asking for the maximum number of copies of a graph $T$ in an $F$-free
graph. In this paper, we determine some of these generalized extremal numbers
and prove supersaturation results for them.
