Connectivity of joins, cohomological quantifier elimination, and an algebraic Toda’s theorem

Abstract

The concept of Vapnik-Chervonenkis (VC) density is pivotal across various mathematical fields, including model theory, discrete geometry, and probability theory. In this paper, we introduce a topological generalization of VC-density. Let Y be a topological space and X , Z (0) , . . . , Z (q−1) be families of subspaces of Y . We define a two parameter family of numbers, vcd p,q X ,Z , which we refer to as the degree p, order q, VC-density of the pair (X , Z = (Z (0) , . . . , Z (q−1) ).The classical notion of VC-density within this topological framework can be recovered by setting p = 0, q = 1. For p = 0, q > 0, we recover Shelah's notion of higher-order VC-density for q-dependent families [37]. Our definition introduces a new notion when p > 0.We examine the properties of vcd p,q X ,Z when the families X and Z (i) are definable in structures with some underlying topology (for instance, the analytic topology over C, the etale site for schemes over arbitrary algebraically closed fields, or the Euclidean topology for o-minimal structures over R). Our main result establishes that that in any model of these theories vcd p,q X ,Z ≤ (p + q) dim X.This result generalizes known VC-density bounds in these structures [3,6,32], extending them in multiple ways, as well as providing a uniform proof paradigm applicable to all of them. We give examples to show that our bounds are optimal. Moreover, our bounds on 0/1-patterns actually goes beyond modeltheoretic contexts: they apply to arbitrary correspondences of schemes with respect to singular, étale, or ℓ-adic cohomology theories. A particular consequence of our results is the extension of the bound on 0/1-patterns for definable families in affine spaces over arbitrary fields, as initially proven in [32], to general schemes.We present combinatorial applications of our higher-degree VC-density bounds, deriving topological analogs of well-known results such as the existence of ε-nets and the fractional Helly theorem. We show that with certain restrictions, these results extend to our higher-degree topological setting.