A combinatorial property of p _κλ

Extract

In a paper on combinatorial properties and large cardinals [2], Jech extended several combinatorial properties of a cardinal κ to analogous properties of the set of all subsets of λ of cardinality less than κ, denoted by “p _κλ”, where λ is any cardinal ≤_κ. We shall consider in this paper one of these properties which is historically rooted in a theorem of Ramsey [10] and in work of Rowbottom [12].

As in [2], define [p _κλ]² = {{x, y}: x, y ∈ p _κλ and x ≠ y}. An unbounded subset A of p _κλ is homogeneous for a function F: [p _κλ]² → 2 if there is a k < 2 so that for all x, y ∈ A with either x ⊊ y or y ⊊ x, F({x, y}) = k. A two-valued measure ü on p _κλ is fine if it is κ-complete and if for all α < λ, ü({x ∈ p _κλ: α ∈ x}) = 1, and ü is normal if, in addition, for every function f: p _κλ → λsuch that ü({x ∈ p _κλ: f(x) ∈ x}) = 1, there is an α < λ such that ü({x ∈ p _κλ: f(x) = α}) = 1. Finally, a fine measure on p _κλ has the partition property if every F: [p _κλ]² → 2 has a homogeneous set of measure one.