Boise Set Theory Seminar, September 2013

Absract: Presenting joint work with Juris Steprāns. This talk is complementary to the earlier talk here.

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. We call a family $\mathcal F$ of infinite subsets of $\mathbb N$ 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 of splitting families, 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 the least cardinality of an $n$-splitting family is the same size for all $n$, we will show that they are in fact distinct notions.

Specifically, we will show that the $n$-splitting relations form a chain in the Borel Tukey ordering on relations of this type. We will also show how to use similar examples to find an infinite antichain in the Borel Tukey ordering.