Submission date: 26 May 2022

Abstract:

Let S be a family of 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 2 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 base case of the definable (p,q)-conjecture in model theory and of the Alon-Kleitman-Matoušek (p,q)-theorem in combinatorics.

Mathematics Subject Classification: 03C45, 52A35

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Full text arXiv 2205.13665: pdf, ps.