Thilo Weinert: Partition relations for linear orders in a non-choice context

Monday, November 18, 2013, 16.30
Seminar room 0.011, Mathematisches Institut, Universität Bonn

Speaker: Thilo Weinert (Universität Bonn)

Title: Partition relations for linear orders in a non-choice context

Abstract: In “Partition Relations for eta_alpha-Sets”, Erdös, Milner and Rado proved, using the Axiom of Choice, three partition relations which amount to the following: -It is possible to partition the triples of any linear order into two classes such that every set of order-type w* + w(the integers) contains a triple in the first class and every quadruple contains a triple in the second class. -It is possible to partition the triples of any linear order into two classes such that every set of order-type w+ w* contains a triple in the first class and every quadruple contains a triple in the second class. -It is possible to partition the triples of any linear order into two classes such that every set of order-type w* + w(the integers) contains a triple in the first class, every set of order-type w+ w* contains a triple in the first class, and every quintuple contains a triple in the second class. They could not tell whether one can replace “quintuple” by “quadruple” in the last statement. Using a structural analysis from ” A Partition Theorem for Perfect Sets” by Blass it is possible to prove analogue statements for a choiceless context for 2x lexicographically ordered for some ordinal x replacing “triple” by “quadruple”, “quadruple” by “quintuple”, and “quintuple” by ” septuple”. This is going to be the topic of the talk.

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