Logic and Set Theory: Sigma_1-partition properties
Philipp Luecke Hausdorff Center, University of Bonn
Howard House 4th Floor Seminar Room
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.