Exact splitting guarantee for hyperplane families

Establish whether Theorem 1 can be strengthened so that every family of finite point sets satisfying the (p,q,α)-property with respect to hyperplanes is α-split by a bounded number of hyperplanes, thereby replacing the theorem’s guarantee β=β(α,d)<α with β=α.

Background

Theorem 1 states that, for fixed α, p, q, and d, a family of finite point sets with the (p,q,α)-property with respect to hyperplanes can be β-split by a bounded number of hyperplanes, where β depends on α and d and is strictly smaller than α.

The concluding remarks identify as unresolved whether the loss in the splitting parameter is necessary. Specifically, they ask whether the same bounded-family conclusion can hold without weakening α to a smaller value.