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

TL;DR

This paper demonstrates that the classical Frankl-Pach upper bound on the size of certain set systems with bounded VC-dimension is not tight for any uniformity, using polynomial methods and structural analysis.

Contribution

It provides a new, general improvement on the Frankl-Pach upper bound applicable for all uniformities, removing previous restrictions.

Findings

Frankl-Pach upper bound is not tight for any uniformity

New polynomial and structural methods improve the bound

Applicable for all $d \\ge 2$ and $n \\ge 2d+2$

Abstract

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 study of this fundamental problem and provided an upper bound $\binom{n}{d}$ via an elegant algebraic proof. Surprisingly, in 2007, Mubayi and Zhao showed that when $n$ is sufficiently large and $d$ is a prime power, the Frankl-Pach upper bound is not tight. They also remarked that their method requires $d$ to be a prime power, and asked for new ideas to improve the Frankl-Pach upper bound without extra assumptions on $n$ and $d$. In this paper, we provide an improvement for any $d\ge 2$ and $n\ge 2d+2$, which demonstrates that the long-standing Frankl-Pach upper bound $\binom{n}{d}$ is not tight for any uniformity. Our proof combines a simple yet powerful polynomial method and structural analysis.