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

Matatyahu Rubin, Saharon Shelah

Research output: Contribution to journalArticlepeer-review

18 Scopus citations

## Abstract

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 Tμ 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 Tμ is a nonempty subset of Tμ closed under initial segments. T≤Tμ means that T is a subtree of Tμ and 〈T, ≤〉 ≊ Tμ. 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{lunate} b}) < λ, then there isT ≤ Tμsuch that Sup({F(α) ∥ α ε{lunate} T}) < λ. Theorem 2 (Rubin). A theorem on large free subtrees. Letλ+ ≤μ, F : Tμ → P(Tμ), for every branch b of Tμ:∥Υ{hooked} {F(α) ∥ α ε{lunate} b} ∥ < λ, and for every α ε{lunate} Tμ and β ε{lunate} F(α), β | ̌α; then there is T ≤ Tμ such that for every α, β ε{lunate} T : β ε{lunate} 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 < μ, and for every κ < μ there is D ∪ Pλ(κ) such that ∥D∥ < μ, and D generates the ideal Pλ(κ) of P(κ). Note that if for every κ < μ κ < 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 : Tμ → P(C) and for every branch b of Tμ: ∥⊂ {F(α) ∥ α ε{lunate} β} ∥ < λ, then there is T ≤ Tμ and a function K : T → Pλ(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).

Original language English 43-81 39 Annals of Pure and Applied Logic 33 C https://doi.org/10.1016/0168-0072(87)90075-3 Published - 1 Jan 1987

• Logic

## Fingerprint

Dive into the research topics of 'Combinatorial problems on trees: Partitions, δ-systems and large free subtrees'. Together they form a unique fingerprint.