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 (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

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

Theorem. Suppose that is -Suslin. Then either

contains a pairwise disjoint perfect subset, i.e., for any two sequences in the subset, for all we have or else

is the union of many graph intersecting -Borel sets, i.e., for any two sequences in any of these sets, there is some such that

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

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

This entry was posted on Friday, April 23rd, 2010 at 8:47 am and is filed under Papers. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

A database of number fields, by Jürgen Klüners and Gunter Malle. (Note this is not the same as the one mentioned in this answer.) The site also provides links to similar databases.

As the other answer indicates, the yes answer to your question is known as the De Bruijn-Erdős theorem. This holds regardless of the size of the graph. The De Bruijn–Erdős theorem is a particular instance of what in combinatorics we call a compactness argument or Rado's selection principle, and its truth can be seen as a consequence of the topological c […]

Every $P_c$ has the size of the reals. For instance, suppose $\sum_n a_n=c$ and start by writing $\mathbb N=A\cup B$ where $\sum_{n\in A}a_n$ converges absolutely (to $a$, say). This is possible because $a_n\to 0$: Let $m_0

Consider a subset $\Omega$ of $\mathbb R$ of size $\aleph_1$ and ordered in type $\omega_1$. (This uses the axiom of choice.) Let $\mathcal F$ be the $\sigma$-algebra generated by the initial segments of $\Omega$ under the well-ordering (so all sets in $\mathcal F$ are countable or co-countable), with the measure that assigns $0$ to the countable sets and $1 […]

You assume $\omega_\alpha\subseteq M$ and $X\in M$ so that $X$ belongs to the transitive collapse of $M$ (because if $\pi$ is the collapsing map, $\pi(X)=\pi[X]=X$. You assume $|M|=\aleph_\alpha$ so that the transitive collapse of $M$ has size $\aleph_\alpha$. Since you also have that this transitive collapse is of the form $L_\beta$ for some $\beta$, it fol […]

No, this is not possible. Dave L. Renfro wrote an excellent historical Essay on nowhere analytic $C^\infty$ functions in two parts (with numerous references). See here: 1 (dated May 9, 2002 6:18 PM), and 2 (dated May 19, 2002 8:29 PM). As indicated in part 1, in Zygmunt Zahorski. Sur l'ensemble des points singuliers d'une fonction d'une variab […]

I don't think you need too much in terms of prerequisites. An excellent reference is MR3616119. Tomkowicz, Grzegorz(PL-CEG2); Wagon, Stan(1-MACA-NDM). The Banach-Tarski paradox. Second edition. With a foreword by Jan Mycielski. Encyclopedia of Mathematics and its Applications, 163. Cambridge University Press, New York, 2016. xviii+348 pp. ISBN: 978-1-10 […]

For the second problem, write $x=-3+x'$ and so on. You have $x'+y'+z'=17$ and $x',\dots$ are nonnegative, a case you know how to solve. You can also solve the first problem this way; now you would set $x=1+x'$, etc.