In this paper we investigate a connection between the growth rates of certain classes of finite structures and a generalization of
V
C
\mathrm {VC}
-dimension called
V
C
ℓ
\mathrm {VC}_{\ell }
-dimension. Let
L
\mathcal {L}
be a finite relational language with maximum arity
r
r
. A hereditary
L
\mathcal {L}
-property is a class of finite
L
\mathcal {L}
-structures closed under isomorphism and substructures. The speed of a hereditary
L
\mathcal {L}
-property
H
\mathcal {H}
is the function which sends
n
n
to
|
H
n
|
|\mathcal {H}_n|
, where
H
n
\mathcal {H}_n
is the set of elements of
H
\mathcal {H}
with universe
{
1
,
…
,
n
}
\{1,\ldots , n\}
. It was previously known that there exists a gap between the fastest possible speed of a hereditary
L
\mathcal {L}
-property and all lower speeds, namely between the speeds
2
Θ
(
n
r
)
2^{\Theta (n^r)}
and
2
o
(
n
r
)
2^{o(n^r)}
. We strengthen this gap by showing that for any hereditary
L
\mathcal {L}
-property
H
\mathcal {H}
, either
|
H
n
|
=
2
Θ
(
n
r
)
|\mathcal {H}_n|=2^{\Theta (n^r)}
or there is
ϵ
>
0
\epsilon >0
such that for all large enough
n
n
,
|
H
n
|
≤
2
n
r
−
ϵ
|\mathcal {H}_n|\leq 2^{n^{r-\epsilon }}
. This improves what was previously known about this gap when
r
≥
3
r\geq 3
. Further, we show this gap can be characterized in terms of
V
C
ℓ
\mathrm {VC}_{\ell }
-dimension, therefore drawing a connection between this finite counting problem and the model theoretic dividing line known as
ℓ
\ell
-dependence.