Combinatorial problems on trees: Partitions, δ-systems and large free subtrees

We prove partition theorems on trees and generalize to a setting of trees the theorems of Erdös and Rado on δ-systems and the theorems of Fodor and Hajnal on free sets. Let μ be an infinite cardinal and T_μ be the tree of finite sequences of ordinals <μ, with the partial ordering of being an initial segment. α≤β denotes that α is an initial segment of β. A subtree of T_μ is a nonempty subset of T_μ closed under initial segments. T≤T_μ means that T is a subtree of T_μ and 〈T, ≤〉 ≊ T_μ. The following are extracts from Section 2, 3 and 4. Theorem 1 (Shelah). A partition theorem. Suppose cf(λ) ≠ cf(μ), F : T_μ → λ, and for every branch b ofT_μ Sup({F(α) ∥ α ε{lunate} b}) < λ, then there isT ≤ T_μsuch that Sup({F(α) ∥ α ε{lunate} T}) < λ. Theorem 2 (Rubin). A theorem on large free subtrees. Letλ⁺ ≤μ, F : T_μ → P(T_μ), for every branch b of T_μ:∥Υ{hooked} {F(α) ∥ α ε{lunate} b} ∥ < λ, and for every α ε{lunate} T_μ and β ε{lunate} F(α), β | ̌α; then there is T ≤ T_μ such that for every α, β ε{lunate} T : β ε{lunate} F(α). Let P_λ(C) denote the ideal in P(C) of all subsets of C whose power is less than λ. Let cov(μ, λ) mean that μ is regular, λ < μ, and for every κ < μ there is D ∪ P_λ(κ) such that ∥D∥ < μ, and D generates the ideal P_λ(κ) of P(κ). Note that if for every κ < μ κ^<λ < cf(μ) = μ, then cov(μ, λ) holds. Let α ∧ β denote the maximal common initial segment of α and β. Theorem 3 (Shelah). A theorem on δ-systems. Suppose Cov(μ, λ) holds, F : T_μ → P(C) and for every branch b of T_μ: ∥⊂ {F(α) ∥ α ε{lunate} β} ∥ < λ, then there is T ≤ T_μ and a function K : T → P_λ(C) such that for every incomparable α, β ε{lunate} T : F(α) ⊃ F(β) ⊆ K(α ∧ β). In 4.12, 4.13, we almost get that K(α) ⊂K(β) = K(α ⊂ β).