A variant of the VC-dimension with applications to depth-3 circuits

We introduce the following variant of the VC-dimension. Given S ⊆ {0,1}n and a positive integer d, we define Ud(S) to be the size of the largest subset I ⊆ [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 Ud dimension is equivalent to a Turán-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 Σk3-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 Σ33-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 Σ23 complexity of the inner product function and all degree-2 polynomials over F2 in general. The question of determining the Σ33 complexity of IP was recently posed by Golovnev, Kulikov, and Williams (ITCS'21).