  We introduce the following variant of the VC-dimension. Given $S \subseteq
\{0, 1\}^n$ and a positive integer $d$, we define $\mathbb{U}_d(S)$ to be the
size of the largest subset $I \subseteq [n]$ such that the projection of $S$ on
every subset of $I$ of size $d$ is the $d$-dimensional cube. We show that
determining the largest cardinality of a set with a given $\mathbb{U}_d$
dimension is equivalent to a Tur\'an-type problem related to the total number
of cliques in a $d$-uniform hypergraph. This allows us to beat the
Sauer--Shelah lemma for this notion of dimension. We use this to obtain several
results on $\Sigma_3^k$-circuits, i.e., depth-$3$ circuits with top gate OR and
bottom fan-in at most $k$:
  * Tight relationship between the number of satisfying assignments of a
$2$-CNF and the dimension of the largest projection accepted by it, thus
improving Paturi, Saks, and Zane (Comput. Complex. '00).
  * Improved $\Sigma_3^3$-circuit lower bounds for affine dispersers for
sublinear dimension. Moreover, we pose a purely hypergraph-theoretic conjecture
under which we get further improvement.
  * We make progress towards settling the $\Sigma_3^2$ complexity of the inner
product function and all degree-$2$ polynomials over $\mathbb{F}_2$ in general.
The question of determining the $\Sigma_3^3$ complexity of IP was recently
posed by Golovnev, Kulikov, and Williams (ITCS'21).
