Extremal Sauer–Shelah problem on Boolean slices

Determine the maximum size of a family of subsets in a Boolean slice with prescribed VC-dimension, including the maximum size of a family of \((d+1)\)-uniform sets having VC-dimension \(d\).

Background

For set families in the full Boolean cube, the Sauer–Shelah–Perles lemma determines the largest possible family size as a function of VC-dimension. The paper states that the corresponding extremal problem on a Boolean slice has not been solved.

The slice version concerns families restricted to sets of a fixed cardinality. The paper specifically points to the problem of determining the maximum size of $(d+1)$-uniform set families whose VC-dimension is $d$, which has been investigated in recent work but is still unresolved.