More definable combinatorics around the first and second uncountable cardinals.

Assume Z F + A D. If is an ordinal and X is a set of ordinals, then [ X ] ∗ is the collection of order-preserving functions f : → X which have uniform cofinality ω and discontinuous everywhere. The weak partition properties on ω 1 and ω 2 yield partition measures on [ ω 1 ] ∗ when < ω 1 and [ ω 2 ] ∗ when < ω 2 . The following almost everywhere continuity properties for functions on partition spaces with respect to these partition measures will be shown. For every < ω 1 and function Φ : [ ω 1 ] → ω 1 , there is a club C ⊆ ω 1 and a ζ < so that for all f , g ∈ [ C ] ∗ , if f ↾ ζ = g ↾ ζ and sup (f) = sup (g) , then Φ (f) = Φ (g). For every < ω 2 and function Φ : [ ω 2 ] → ω 2 , there is an ω -club C ⊆ ω 2 and a ζ < so that for all f , g ∈ [ C ] ∗ , if f ↾ ζ = g ↾ ζ and sup (f) = sup (g) , then Φ (f) = Φ (g). The previous two continuity results will be used to distinguish the cardinalities of some important subsets of (ω 2) : | [ ω 1 ] ω | < | [ ω 1 ] < ω 1 |. | [ ω 2 ] ω | < | [ ω 2 ] < ω 1 | < | [ ω 2 ] ω 1 | < | [ ω 2 ] < ω 2 |. ¬ (| [ ω 1 ] < ω 1 | ≤ | [ ω 2 ] ω |). ¬ (| [ ω 1 ] ω 1 | ≤ | [ ω 2 ] < ω 1 |). It will also be shown that [ ω 1 ] ω has the Jónsson property: For every Φ : < ω ([ ω 1 ] ω) → [ ω 1 ] ω , there is an X ⊆ [ ω 1 ] ω with | X | = | [ ω 1 ] ω | so that Φ [ < ω X ] ≠ [ ω 1 ] ω . [ABSTRACT FROM AUTHOR]