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Choiceless Ramsey Theory of Linear Orders

Lücke, Philipp ; Schlicht, Philipp ; Weinert, Thilo

Order (Dordrecht), 2017-11, Vol.34 (3), p.369-418 [Periódico revisado por pares]

Dordrecht: Springer Netherlands

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  • Título:
    Choiceless Ramsey Theory of Linear Orders
  • Autor: Lücke, Philipp ; Schlicht, Philipp ; Weinert, Thilo
  • Assuntos: Algebra ; Coloring ; Discrete Mathematics ; Lattices ; Mathematics ; Mathematics and Statistics ; Number theory ; Order ; Ordered Algebraic Structures ; Partitions (mathematics) ; Series (mathematics)
  • É parte de: Order (Dordrecht), 2017-11, Vol.34 (3), p.369-418
  • Descrição: 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,< l e x 〉, where α is an infinite ordinal and < l e x 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,< l e x 〉 from the property of Baire for all subsets of ω 2 and show that the relation 〈 κ 2 , < lex 〉 → ( 〈 κ 2 , < lex 〉 ) 2 2 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,< l e x 〉. We combine the positive and negative results to completely classify which of the partition relations 〈 ω 2 , < lex 〉 → ( ∨ ν < λ K ν , ∨ ν < μ M ν ) m for linear orders K ν , M ν and m ≤4 and 〈 ω 2,< l e x 〉→( K , M ) n for linear orders K , M and natural numbers n are consistent.
  • Editor: Dordrecht: Springer Netherlands
  • Idioma: Inglês
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  • Realização: ABCD-USP USP   Logos de Redes Sociais: Freepik

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