Philipp Lücke: Sigma_1-partition properties

Tuesday, February 14, 2017, 15.00
Howard House 4th Floor Seminar Room

Speaker: Philipp Lücke (Hausdorff Centre, University of Bonn)

Title: Sigma_1-partition properties

Abstract:

We consider colourings of the set of pairs of countable ordinals with two colours that are definable by Sigma_1-formulas that only use the first uncountable cardinal omega_1 and real numbers as parameters. We present results showing that the existence of a measurable cardinal above a Woodin cardinal implies that uncountable homogeneous sets exist for all such colourings. In contrast, a failure of this partition property is compatible with the existence of a single Woodin cardinal. Finally, we show that similar definable partition properties can hold for large cardinals that are not weakly compact; e.g. stationary limits of omega_1-iterable cardinals.