A tree \(T\) of height \(\omega_1\) with no uncountable branch is special if there is a function \(f:T \to \omega\) which is injective on chains. It's well known that under \(\mathrm{MA} + \neg \mathrm{CH}\) every tree of height \(\omega_1\) with no uncountable branch of size less than the continuum is special, while in \(\mathrm{ZFC}\) one can construct a non-special tree of height \(\omega_1\) with no uncountable branch. At the same time there may be a Souslin tree while the continuum is as large as you like thus providing a model with a small non-special tree. These facts together suggest a new cardinal characteristic, the special tree number, denote \(\mathfrak{st}\): the least size of a tree of height \(\omega_1\) with no uncountable branch which is not special. By what was observed above, \(\mathrm{MA} + \neg \mathrm{CH}\) implies that \(\mathfrak{st} = 2^{\aleph_0}\) while it is consistent that \(\mathfrak{st} < 2^{\aleph_0}\) with the latter arbitrarily large.

In this talk we will introduce the basic properties of \(\mathfrak{st}\) and prove in particular that it is consistent on the one hand that \(\mathfrak{st}\) is \(\aleph_1\) while essentially all well-studied cardinal characteristics are arbitrarily large and on the other hand it is consistent that for any regular \(\kappa\) we have \(\mathfrak{a} = {\rm non}(\mathcal M) = \aleph_1 < \mathfrak{st} = {\rm cov}(\mathcal M) = 2^{\aleph_0} = \kappa\). In other words, \(\mathfrak{st}\) is independent of the lefthand side of Cichoń's diagram, \(\mathfrak{p}\) and \(\mathfrak{a}\). The latter model involves a careful analysis of reals added by the standard ccc forcing to specialize trees, which may be of independent interest.

This is a relatively new investigation and there are many open questions I hope to discuss as well, time permitting.

This talk will be given in mixed mode, in person as well as via Zoom.

If you want to attend in person, please be aware of the fact that you will be required to show proof of your COVID-19 "2.5G" status (vaccinated, recovered, PCR tested) upon entry of the buildings, or during sporadic random checks in the seminar rooms. During the lectures we will also pass around an attendance sheet to facilitate contact tracing. (According to the regulations, this form will be kept for 28 days and destroyed thereafter.)