Sam Coskey, a postdoc at the Fields Institute and York University, speaks on "Borel morphisms and splitting families".

Abstract:

We will define and discuss the notion of a Tukey morphism between cardinal invariants of the continuum. These are important because the existence of such a map between x and y implies that x≥y. However,
in practice one is more concerned with true inequalities, that is, inequalities which hold in all models of ZFC. For this reason, Blass proposed that we consider the Borel Tukey morphisms, which are absolute to forcing extensions.

After setting up the basics, I will discuss the similarities and differences between the usual ordering and the Borel Tukey orderings on a modest collection of classical combinatorial cardinal invariants.

In the last part of the talk, I will focus on the splitting number and some generalizations. Namely, F is said to be an n-splitting family if for every sequence of infinite sets B1,...,Bn there exists A∈F which splits them all. Although each of these types of families determine the same cardinal invariant (that is, the smallest n-splitting family has the same size as the smallest splitting family), we will show that the n-splitting relations are properly increasing in the Borel Tukey order.

This is joint work with Juris Steprāns. It was motivated by a question of Blass (solved by Mildenberger), which asked whether the same result holds for reaping families and its generalizations to n-reaping families.

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