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Definable $(ω, 2)$-theorem for families with VC-codensity less than $2$

Pablo Andújar Guerrero

TL;DR

Let $\mathcal{S}$ be a family of sets with dual shatter function $\pi^*_{\mathcal{S}}(n)\in o(n^2)$ (VC-codensity $<2$). If $\mathcal{S}$ has the $(\omega,2)$-property, the paper shows that $\mathcal{S}$ can be partitioned into finitely many subfamilies each with the finite intersection property, and that these subfamilies can be chosen definable when $\mathcal{S}$ is definable in a structure. The approach combines model-theoretic notions of definability with combinatorial bounds on the dual shatter function, avoiding reliance on the standard $(p,q)$-theorem proofs. This yields a definable $(\omega,2)$-theorem for $q=2$, strengthening the definable $(p,2)$-conjecture in NIP settings and connecting VC-density with finite-intersection decompositions. The results illuminate how low VC-density constraints enforce controllable definable partitions, with potential implications for tame geometry and related areas.

Abstract

Let $\mathcal{S}$ be a family of sets with VC-codensity less than $2$. We prove that, if $\mathcal{S}$ has the $(ω, 2)$-property (for any infinitely many sets in $\mathcal{S}$, at least $2$ among them intersect), then $\mathcal{S}$ can be partitioned into finitely many subfamilies, each with the finite intersection property. If $\mathcal{S}$ is definable in some first-order structure, then these subfamilies can be chosen definable too. This is a strengthening of the case $q=2$ of the definable $(p,q)$- conjecture in model theory and of the Alon-Kleitman-Matoušek $(p,q)$-theorem in combinatorics.

Definable $(ω, 2)$-theorem for families with VC-codensity less than $2$

TL;DR

Let be a family of sets with dual shatter function (VC-codensity ). If has the -property, the paper shows that can be partitioned into finitely many subfamilies each with the finite intersection property, and that these subfamilies can be chosen definable when is definable in a structure. The approach combines model-theoretic notions of definability with combinatorial bounds on the dual shatter function, avoiding reliance on the standard -theorem proofs. This yields a definable -theorem for , strengthening the definable -conjecture in NIP settings and connecting VC-density with finite-intersection decompositions. The results illuminate how low VC-density constraints enforce controllable definable partitions, with potential implications for tame geometry and related areas.

Abstract

Let be a family of sets with VC-codensity less than . We prove that, if has the -property (for any infinitely many sets in , at least among them intersect), then can be partitioned into finitely many subfamilies, each with the finite intersection property. If is definable in some first-order structure, then these subfamilies can be chosen definable too. This is a strengthening of the case of the definable - conjecture in model theory and of the Alon-Kleitman-Matoušek -theorem in combinatorics.
Paper Structure (5 sections, 7 theorems, 28 equations)

This paper contains 5 sections, 7 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Preliminary results
  5. Proof of the main result

Key Result

Theorem 1

Let $q\geq 2$ be an integer and $\mathcal{S}$ be a family of sets whose dual shatter function satisfies $\pi_{\mathcal{S}}^*(n)\in o(n^q)$ (that is, $\lim_{n\rightarrow \infty} \pi_{\mathcal{S}}^*(n)/n^q =0$). For any integer $p\geq q$, there exists some $m<\omega$ such that, if $\mathcal{F}$ is a s

Theorems & Definitions (16)

  • Theorem 1: Alon-Kleitman-Matoušek $(p,q)$-theorem
  • Conjecture 2: Definable $(p,q)$-conjecture
  • Theorem 3: Definable $(\omega,2)$-theorem
  • Corollary 4: $(\omega,2)$-theorem
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 6 more