Dear all,

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.

Best,

David