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 -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 of sets and let . We say that is intersecting iff any two members of share at least an element of . In our paper we are interested in how definability conditions restrict the size of intersecting families. In particular we show that, even if and are infinite, if all the members of are finite, then from we can define a finite subset of .
Let be such that there is a set in of size at most . Then our arguments also show that there is a finite intersecting family definable from , and we find upper bounds on how large such is.
Say that is -minimal iff is intersecting and any intersecting definable from is at least as large as itself. It follows that there is a finite bound on how large such a family can be.
Question. Is a (strictly) increasing function?
We can show that and a few similar results, but whether 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.]
[…] 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 […]