Definable (ω,2)-Theorem for Families with VC-Codensity less than 2

Let S be a family of nonempty sets with VC-codensity less than 2. We prove that, if S has the (ω,2)-property (for any infinitely many sets in S, at least two among them intersect), then S can be partitioned into finitely many subfamilies, each with the finite intersection property. If 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 the Alon-Kleitman-Matou¿ek (p,q)-theorem in combinatorics.