Maxmum Size of a Uniform Family with Bounded VC-dimension

TL;DR

This paper improves the upper bound on the size of uniform families with bounded VC-dimension, narrowing the gap between known lower and upper bounds and advancing understanding in combinatorial set theory.

Contribution

It further reduces the upper bound on the maximum size of such families to asymptotically match the known lower bound, refining previous results.

Findings

Upper bound reduced to inom{n-1}{d} + O(n^{d-2})

Asymptotic match with the lower bound inom{n-1}{d}+inom{n-4}{d-2}

Progress towards resolving the Frankl--Pach conjecture

Abstract

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 $\binom{n-1}{d}+\binom{n-4}{d-2}$.