Defining Non-empty Small Sets from Families of Infinite Sets

We describe circumstances under which non-empty small sets can be defined from families of infinite sets. In the process, we establish generalizations of Mansfield’s perfect set theorem for κ-Souslin sets and the Lusin-Novikov uniformization theorem. An intersecting family is a collection of sets, any two of which have non-empty intersection. These have been the subject of much study in combinatorics, and have also recently come up in descriptive set theory. In particular, while [CCM07] focused on σ-ideals associated with countable Borel equivalence relations, the main result there depended on the simple but surprising observation that for all sets X and all nonempty intersecting families A of finite subsets of X, a non-empty finite subset of X is definable from A . This observation was generalized and strengthened in [CCCM09], where quantitative analogs were obtained for non-empty families of non-empty sets that do not contain infinite pairwise disjoint subfamilies. To be precise, let [X] + denote the family of all non-empty subsets of X whose cardinality is at most κ, and let L denote the signature consisting of a unary relation symbol ̇ A and a binary relation symbol ∈̇. Associated with each cardinal κ, set X, and family A ⊆ [X] + is the L-structure MA = (X ∪ [X] + ,A ,∈ ∩ (X × [X] ≤κ + )). We do not specify κ in our notation as it will be clear from context. Observe that both X and [X] + are definable in MA . Theorem 1. There is a disjunction of first-order L-formulae θ(x) with the property that if k ∈ ω, X is a set, and A ⊆ [X] + is a non-empty family that does not have an infinite pairwise disjoint subfamily, then {x |MA |= θ(x)} is a non-empty finite subset of X. 2010 Mathematics Subject Classification. Primary 03E15; Secondary 05B30.