2011 seminar talk: Conjectures of Rado and Chang and Special $\aleph_2$-Aronszajn trees

Talk held by Victor Torres (KGRC) at the KGRC seminar on 2011-04-14.

Abstract

We recall that Rado's Conjecture (RC) is the statement that every tree T that is not decomposable into countably many antichains contains a subtree of cardinality \(\aleph_1\) with the same property. Todorcevic has shown the consistency of this statement relative to the consistency of the existence of a strongly compact cardinal. Moreover he has shown that RC is consistent with CH as well as consistent with the negation of CH. Also, he has shown that RC has many interesting consequences, including the continuum is at most \(\aleph_2\), CC, SCH, the negation of \(\Box_\kappa\) for all \(\kappa > \omega\), etc. This result is the first in the series which will study the effect of RC or one of its consequences to weaker form of square sequences. We will show that Jensen's weak square principle \(\Box_{\omega_1}^*\) is equivalent to CH if we assume either Rado's conjecture or its consequence, the strong Chang's Conjecture.