## Abstract

Motivated by work of Erdős, Milner and Rado, we investigate symmetric and asymmetric partition relations for linear orders without the axiom of choice. The relations state the existence of a subset in one of finitely many given order types that is homogeneous for a given colouring of the finite subsets of a fixed size of a linear order. We mainly study the linear orders 〈^{α}2,<_{lex}〉, where α is an infinite ordinal and <_{lex} is the lexicographical order. We first obtain the consistency of several partition relations that are incompatible with the axiom of choice. For instance we derive partition relations for 〈^{ω}2,<_{lex}〉 from the property of Baire for all subsets of ^{ω}2 and show that the relation 〈κ2,<lex〉→(〈κ2,<lex〉)22 is consistent for uncountable regular cardinals κ with κ^{<κ} = κ. We then prove a series of negative partition relations with finite exponents for the linear orders 〈^{α}2,<_{lex}〉. We combine the positive and negative results to completely classify which of the partition relations 〈ω2,<lex〉→(Kν,Mν)m for linear orders K_{ν},M_{ν} and m≤4 and 〈^{ω}2,<_{lex}〉→(K,M)^{n} for linear orders K,M and natural numbers n are consistent.

Original language | English |
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Pages (from-to) | 369-418 |

Number of pages | 50 |

Journal | Order |

Volume | 34 |

Issue number | 3 |

DOIs | |

State | Published - 1 Nov 2017 |

Externally published | Yes |

## Keywords

- Axiom of choice
- Axiom of determinacy
- Linear orders
- Partition relations
- Ramsey theory

## ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology
- Computational Theory and Mathematics