Determine the maximum size of a uniform set system with bounded VC-dimension

Determine the maximum size of a (d+1)-uniform set system on [n] having VC-dimension at most d.

Background

The paper studies the extremal function for (d+1)-uniform set systems with VC-dimension at most d. The non-uniform analogue is completely determined by the SauerShelahPerles lemma, but the uniform problem remains unresolved. Frankl and Pach proved the upper bound \binom{n}{d}, while Ahlswede and Khachatrian constructed examples of size \binom{n-1}{d}+\binom{n-4}{d-2}. The main theorem improves the upper bound substantially but does not determine the exact maximum.