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 [9] and the Alon–Kleitman–Matoušek (p,q) -theorem in combinatorics [6].