On a conjecture of Terry and Wolf

TL;DR

This paper establishes bounds on the VC-dimension of a specific subset of finite fields, confirming a conjecture and simplifying existing proofs related to combinatorial and model-theoretic properties.

Contribution

It provides new bounds on the VC-dimension of the quadratic Green-Sanders example, confirming a conjecture and offering a simplified proof for the linear case.

Findings

VC_2-dimension of quadratic Green-Sanders example is between 3 and 501.

Confirmed the upper bound conjecture of Terry and Wolf.

Simplified proof that the linear Green-Sanders example has VC-dimension at most 3.

Abstract

This paper shows that the $\mathrm{VC}_2$-dimension of a subset of $\mathbb{F}_p^n$ known as the 'quadratic Green-Sanders example' is at least 3 and at most 501. The upper bound confirms a conjecture of Terry and Wolf, who introduced this set in their recent work concerning strengthenings of the higher-order arithmetic regularity lemma under certain model-theoretic tameness assumptions. Additionally, the paper presents a simplified proof that the (linear) Green-Sanders example, which has its roots in Ramsey theory, has $\mathrm{VC}$-dimension at most 3.