Rado´s Conjecture, the Tree Property, Square Principles and Topology

In this proposal we would like to extend the applications and properties of a very weak form of the compactness through a combinatorial principle called Rado`s Conjecture, RC, introduced by Richard Rado some thirty years ago. It is a compactness principle for the chromatic number of intersection graphs on families of intervals of linearly ordered sets. It states that if such a graph is not countably chromatic then it contains a subgraph of size the first uncountable cardinal which is also not countably chromatic. In 1983, Stevo Todorcevic showed that it is consistent but independent from the traditional axioms of Set Theory. Todorcevic proved in 1993 that Rado`s Conjecture has similar consequences to other principles such as Martin Maximum: it implies the Singular Cardinal Hypothesis, the negation of Jensen`s square principle, it bounds the continuum to the second uncountable cardinal, etc. However, Rado`s Conjecture is not compatible even with one of the first forcing axioms ever considered, Martin`s Axiom. More recently, our recent work with Todorcevic, and the one by Hiroshi Sakai has established a general and rather complete picture of the relation between Rado`s Conjecture and weak forms of square principles. We would like to study now the relationship between Rado`s Conjecture and Todorcevic`s square principle in a two cardinal version, and to get a general picture as the ones with Schimmerling`s two cardinal version of the square principle. Recently we have obtained some partial results, and this proposal is to continue our research in this direction. In a joint work with Liuzhen Wu, we have shown that Rado`s Conjecture, together with the negation of the continuum hypothesis, can give us the Tree Property for the second uncountable cardinal, and and even more, we can get the Strong Tree Property for such cardinal. A relevant question that we would like to study in this proposal is if Rado`s Conjecture with the negation of the continuum hypothesis is strong enough to give us the Super Tree Property for the second uncountable cardinal. Also, this proposal is in the sense to explore new horizons of this Conjecture. Even if Martin`s Axiom is not compatible with Rado`s Conjecture,we would like to verify if there is a weak form of MA consistent with RC. Recently, it has been proven by Fuchino that Rado`s Conjecture implies the Fodor-Type Reflection Principle, which is equivalent to many topological reflecting principles. From here, there are natural questions that arise. For example: Is Rado`s Conjecture itself equivalent to a topological reflecting principle? This proposal is an attempt to answer these and other related questions.

We studied new scopes of a rather new combinatorial principle called Rados Conjecture (RC). This principle has two particularities in the context of modern Set Theory. On one hand, it is independent from the common axioms of mathematics. On the other hand, it is not compatible with more recent and commonly used axioms, such as forcing axioms. So it is quite unique in the realm of Set Theory. However, the Conjecture of Rado has similar consequences as these mentioned forcing axioms. These consequences involve that the size of the real line cannot be larger than the second uncountable cardinal, the Singular Cardinal Hypothesis, the negation of the square principle, etc.In this project we obtained new results with several coauthors from universities of China, France, Japan and Austria involving RC and its relation with 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 observed to which extent this may hold true and where we can find some limitations. These project left some interesting open problems and possible new directions.