Time:  3:30pm  4:30 pm 
Room: 
Wean Hall 8220

Speaker: 
Jing Zhang Department of Mathematical Sciences CMU 
Title: 
Rado's Conjecture and its Baire Version

Abstract: 
Rado's Conjecture is a reflection/compactness principle formulated by Todorčević, who also showed its consistency relative to the existence of strongly compact cardinals. One of its equivalent forms asserts that any nonspecial tree of height ω_{1} has a nonspecial subtree of size less or equal to ℵ_{1}. Although it is incompatible with Martin's Axiom, Rado's Conjecture turns out to imply a lot of consequences of forcing axioms, for example Strong Chang's Conjecture, failure of square principles, the semistationary reflection principle, the Singular Cardinal Hypothesis etcetera. In fact, almost all known consequences of Rado's Conjecture are consequences of a weaker statement, the Baire version of it which asserts any Baire tree of height ω_{1} has a nonspecial subtree of size less or equal to ℵ_{1}. We will show that in the forcing extension by countable support iteration of Sacks forcing of strongly compact length, the Baire version of Rado's Conjecture holds. Using a classical Mitchell style model, we show Rado's conjecture along with notCH does not imply ω_{2} has the super tree property, answering a question by TorresPérez and Wu. We will also see that in general the Baire version of Rado's Conjecture does not imply Rado's Conjecture. 