Borel morphisms and splitting families
Michigan logic seminar, Ann Arbor, November 2011
Description: This talk will begin with the material from the earlier talk posted here. After introducing the background we will focus on just the so-called splitting relation and some of its generalizations. Again, it is joint with Juris Steprāns.
What is the splitting relation? If $A$ and $B$ are infinite subsets of $\mathbb N$, we say that $A$ splits $B$ if both $A\cap B$ and $A^c\cap B$ are infinite. In other words, if we think of $\mathbb N$ as being partitioned into $A$ and $A^c$, then $B$ sits nontrivially on both sides. This is a pretty natural relationship between subsets of $\mathbb N$, and it arises now and again in set theory. As usual, we are interested in families that are dominating with respect to this relation:
Definition. A family $\mathcal F$ of infinite subsets of $\mathbb N$ is said to be a splitting family if for every infinite set $B$ there is $A\in\mathcal F$ such that $A$ splits $B$.
In this talk, we consider some natural generalizations, namely, $\mathcal F$ is said to be an $n$-splitting family if for every sequence of infinite sets $B_1,\ldots,B_n$ there exists $A\in\mathcal F$ which splits them all. Although each of these relations determines the same cardinal invariant (that is, the smallest $n$-splitting family has the same size as the smallest splitting family), we will show that they are distinct notions.
Moreover, we will show that whenever $n<m$, there cannot be a Borel function which carries $n$-splitting families to $m$-splitting families. In fact, we will show that the $n$-splitting relations are properly increasing in the Borel Tukey order.
his study 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.