The seminar meets on Wednesday October 3rd at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.
Program: Victor Torres Perez — Combinatorial Principles without MA
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 ℵ_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 ℵ_2, the
negation of Box(θ) for every regular θ ≥ ω_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(ℵ_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.
For example, we will also discuss some recent results regarding the
P-ideal dichotomy (which can be consistent with the negation of MA(ℵ_1))
and square principles.