Let $S \subset \{-1,1\}^n$. For a subset $A \subset [n]$ let $P_A$ denote the coordinate projection operator on S; in other words let $P_A(S)$ be the coordinate projection of $S$ onto the coordinates indexed by $A$. Now assume that $|P_A(S)| > 2^{c |A| }$ for every A and some fixed positive constant $c$.
Does it follow that $[n]$ can be partitioned into a finite number (say, $M := M(c)$) of sets $B_1,B_2,\ldots,B_M$ such that $|P_{B_i}(S)| = 2^{|B_i|}$?
Of course, the standard Sauer-Shelah lemma gives a single set B of size proportional to n satisfying the conclusion. Thus one can iterate the Sauer-Shelah lemma to obtain a decomposition into $\log n$ such sets. The question is if the log n can be removed (or at least reduced).
This has some (at least superficial) similarities with Weaver's formulation of Kadison-Singer or the Rado-Horn Theorem.
