Splitting relations

Boise Set Theory Seminar, Boise, September 2013

Abstract: 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.

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