Maxmum Size of a Uniform Family with Bounded VC-dimension

In 1984, Frankl and Pach proved that, for positive integers $n$ and $d$, the maximum size of a $(d+1)$-uniform set family $\mathcal{F}$ on an $n$-element set with VC-dimension at most $d$ is at most ${n\choose d}$; and they suspected that ${n\choose d}$ could be replaced by ${n-1\choose d}$, which would generalize the famous Erd\H{o}s-Ko-Rado theorem and was mentioned by Erd\H{o}s as Frankl--Pach conjecture. However, Ahlswede and Khachatrian in 1997 constructed $(d+1)$-uniform families on an $n$-element set with VC-dimension at most $d$ and size exactly $\binom{n-1}{d}+\binom{n-4}{d-2}$, and Mubayi and Zhao in 2007 constructed more such families. It has since been an open question to narrow the gap between the lower bound $\binom{n-1}{d}+\binom{n-4}{d-2}$ and the upper bound ${n\choose d}$. In a recent breakthrough, Chao, Xu, Yip, and Zhang reduced the upper bound $\binom{n }{d}$ to $\binom{n-1}{d}+O( n^{d-1-\frac{1}{4d-2}})$. In this paper, we further reduce the upper bound to $\binom{n-1}{d} + O(n^{d-2})$, asymptotically matching the lower bound $\binom{n-1}{d}+\binom{n-4}{d-2}$.