On a conjecture of Terry and Wolf

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.

Corrections to the case-analysis in the lower-bound section, and some typos fixed