## 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 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_{μ}such 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).

Original language | English |
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Pages (from-to) | 43-81 |

Number of pages | 39 |

Journal | Annals of Pure and Applied Logic |

Volume | 33 |

Issue number | C |

DOIs | |

State | Published - 1 Jan 1987 |

## ASJC Scopus subject areas

- Logic