## Intersecting families and definability

I have made some changes to my webpage and now my papers can be found here, on a page in this blog.

I have updated my paper Defining small sets from $\kappa$-cc families of sets, joint with Clemens, Conley and Miller. I would like to mention one of the questions we leave open.

Consider a family ${\mathcal A}$ of sets and let $X=\bigcup{\mathcal A}$. We say that ${\mathcal A}$ is intersecting iff any two members of ${\mathcal A}$ share at least an element of $X$. In our paper we are interested in how definability conditions restrict the size of intersecting families. In particular we show that, even if ${\mathcal A}$ and $X$ are infinite, if all the members of ${\mathcal A}$ are finite, then from ${\mathcal A}$ we can define a finite subset of $X$.

Let $n$ be such that there is a set in ${\mathcal A}$ of size at most $n$. Then our arguments also show that there is a finite intersecting family ${\mathcal A}'\subseteq[X]^{\le n}$ definable from ${\mathcal A}$, and we find upper bounds on how large such ${\mathcal A}'$ is.

Say that ${\mathcal A}$ is $n$-minimal iff ${\mathcal A}\subseteq[X]^{\le n}$ is intersecting and any intersecting ${\mathcal A}'\subseteq[X]^{\le n}$ definable from ${\mathcal A}$ is at least as large as ${\mathcal A}$ itself. It follows that there is a finite bound $\psi(n)$ on how large such a family ${\mathcal A}$ can be.

Question. Is $\psi$ a (strictly) increasing function?

We can show that $\psi(n)<\psi(2n)$ and a few similar results, but whether $\psi$ is monotone is still open.

[Update, April 23, 2010: To reflect a few recent changes that resulted in a companion paper dealing with infinite sets, we have changed the title to Defining non-empty small sets from families of finite sets.]