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

Abstract We prove partition theorems on trees and generalize to a setting of trees the theorems of Erdos and Rado on δ-systems and the theorems of Fodor and Hajnal on free sets. Let μ be an infinite cardinal and Tμ be the tree of finite sequences of ordinals 〈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 : Tμ → λ, and for every branch b ofTμ Sup({F(α) ∥ α ϵ b}) Theorem 2 (Rubin). A theorem on large free subtrees. Letλ+ ⩽μ, F : Tμ → P(Tμ), for every branch b of Tμ:∥ϒ {F(α) ∥ α ϵ b} ∥ | α; then there is T ⩽ Tμ such that for every α, β ϵ T : β ϵ F(α). Let Pλ(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 Theorem 3 (Shelah). A theorem on δ-systems. Suppose Cov(μ, λ) holds, F : Tμ → P(C) and for every branch b of Tμ: ∥⊂ {F(α) ∥ α ϵ β} ∥ In 4.12, 4.13, we almost get that K(α) ⊂K(β) = K(α ⊂ β).