## Defining non-empty small sets from families of infinite sets

John Clemens, Clinton Conley, Benjamin Miller, and I have just submitted to Fundamenta Mathematicae the second part of a paper I mentioned here a while ago. The preprint is available at my papers page.

This paper deals with families of infinite sets, typically given in some definable way in the descriptive set theoretic sense. In analogy with the results in our companion paper, that deals with finite sets, we show here how in a first-order way we can define intersecting families, and how we can extract from them small sets, at least under reasonable restrictions.

These results have been in the works for a while, but it is only very recently that we have been able to state them in their current generality, thanks to recent advances due to Ben Miller. In the mean time, Richard Ketchersid and I have proved some results in models of ${\sf AD}^+$ (continuing the work reported here) that have also allowed the four of us to find a few extensions of the results in this paper to models of ${\sf AD}^+.$

Here is a sample result, a generalization of the perfect set theorem:

Theorem. Suppose that $A\subseteq\omega^\omega$ is $\kappa$-Suslin. Then either

1. $A$ contains a pairwise disjoint perfect subset, i.e., for any two sequences $f,g$ in the subset, for all $n$ we have $f(n)\ne g(n),$ or else
2. $A$ is the union of $\kappa$ many graph intersecting $\kappa^+$-Borel sets, i.e.,  for any two sequences $f,g$ in any of these sets, there is some $n$ such that $f(n)=g(n).$

Here, a set is $\kappa$-Borel iff it comes from basic open sets by taking unions and complements, where we now allow the unions to have size $\kappa$ rather than just countable.

An interesting side effect of our arguments is that the results are provable in ${\sf ZF}.$ We also consider a few extensions that use the axiom of choice, but have decided to indicate explicitly whenever choice is required.