DEFINABLE -THEOREM FOR FAMILIES WITH VC-CODENSITY LESS THAN

Abstract

Abstract Let $\mathcal {S}$ be a family of nonempty 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 two 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 [9] and the Alon–Kleitman–Matoušek $(p,q)$ -theorem in combinatorics [6].