# Víctor Torres-Pérez: Rado’s Conjecture, an alternative to forcing axioms?

KGRC Research Seminar – 2017‑05‑04 at 4pm

Speaker: Víctor Torres-Pérez (TU Wien)

Abstract: Rado’s Conjecture (RC) in the formulation of Todorcevic 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.

Todorcevic also showed that RC implies the Singular Cardinal Hypothesis,
a strong form of Chang’s Conjecture, the continuum is at most $\aleph_2$,
the negation of $\Box(\theta)$ for every regular $\theta\geq\omega_2$,
etc. These implications are very similar to the ones obtained from traditional
forcing axioms such as MM or PFA. However, RC is incompatible even with
$MA(\aleph_1)$.

In this talk we will take the opportunity to give an overview of our
results with different coauthors obtained in the last few years together
with recent ones, involving RC, certain weak square principles and
instances of tree properties. These new implications seem to continue
suggesting that RC is a good alternative to forcing axioms. We will discuss
to which extent this may hold true and where we can find some limitations.
We will end the talk with some open problems and possible new directions.