Let
k
r
(
n
,
m
)
k_{r}\left (n,m\right )
denote the minimum number of
r
r
-cliques in graphs with
n
n
vertices and
m
m
edges. For
r
=
3
,
4
r=3,4
we give a lower bound on
k
r
(
n
,
m
)
k_{r}\left (n,m\right )
that approximates
k
r
(
n
,
m
)
k_{r}\left (n,m\right )
with an error smaller than
n
r
/
(
n
2
−
2
m
)
.
n^{r}/\left (n^{2}-2m\right ).
The solution is based on a constraint minimization of certain multilinear forms. Our proof combines a combinatorial strategy with extensive analytical arguments.