Some model-theoretic consequences of high-arity uniform convergence, part I

TL;DR

This paper demonstrates that certain complex families of sets in Euclidean spaces can be uniformly approximately definable within o-minimal structures, even if they lack bounded VC-dimension or definability.

Contribution

It establishes model-theoretic consequences of high-arity uniform convergence, expanding understanding of definability beyond traditional VC-dimension constraints.

Findings

Families of sets in $ eal^2$ are uniformly approximately definable in o-minimal structures.

Such families can be neither definable nor have bounded VC-dimension.

The results connect model theory with geometric set families in Euclidean spaces.

Abstract

We show that certain families of sets in $\mathbb{R}^2$ (or $\mathbb{R}^n$) which are neither definable nor have bounded VC-dimension are nonetheless uniformly approximately definable in the real field, an o-minimal structure.