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

TL;DR

This paper proves that families of sets with VC-codensity less than 2 and the $(, 2)$-property can be finitely partitioned into subfamilies with the finite intersection property, extending key conjectures in model theory and combinatorics.

Contribution

It establishes a definable version of the $(, 2)$-theorem for such families, strengthening existing conjectures in model theory and combinatorics.

Findings

Families with VC-codensity < 2 and the $(, 2)$-property can be finitely partitioned.

Partitions can be chosen to be definable if the family is definable.

Extends the $(p,q)$-conjecture and the Alon-Kleitman-Matousek $(p,q)$-theorem.

Abstract

Let $\mathcal{S}$ be a family of sets with VC-codensity less than $2$. We prove that, if $\mathcal{S}$ has the $(\omega, 2)$-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-Matou\v{s}ek $(p,q)$-theorem in combinatorics.