Thursday, July 20, 2017, 10:30–12.00

Main Lecture Hall , Alfréd Rényi Institute of Mathematics

**Abstract:** We study the uncountable version of a natural variant of the Open Coloring Axiom. More concretely, suppose that $\kappa$ is an uncountable cardinal such that $\kappa^{<\kappa}=\kappa$ and X is a subset of the generalized Baire space $\kappa^\kappa$ (the space of functions from $\kappa$ to $\kappa$ equipped with the bounded topology). Let OCA*(X) denote the following statement: for every partition of $[X]^2$ as the union of an open set R and a closed set S, either X is a union of $\kappa$ many S-homogeneous sets, or there exists a $\kappa$-perfect R-homogeneous set. We show that after Lévy-collapsing an inaccessible $\lambda>\kappa$ to $\kappa^+$, OCA*(X) holds for all $\kappa$-analytic subsets X of $\kappa^\kappa$. Furthermore, the Silver dichotomy for ${\Sigma}^0_2(\kappa)$ equivalence relations on $\kappa$-analytic subsets also holds in this model. Thus, both of the above statements are equiconsistent with the existence of an inaccessible $\lambda>\kappa$. We also examine games related to the above partition properties.