The Frankl-Pach upper bound is not tight for any uniformity

Gennian Ge, Zixiang Xu, Chi Hoi Yip +2

For any positive integers $n\ge d+1\ge 3$, what is the maximum size of a $(d+1)$-uniform set system in $[n]$ with VC-dimension at most $d$? In 1984, Frankl and Pach initiated the s…