Forcing Axioms and Compactness Principles without MA

Content of research project In this project, we would like to study the properties of two relative new axioms: YPFA and Rado`s Conjecture (RC). Both principles are independent of ZFC, the traditional axioms of Set Theory. These principles have already some interesting and deep implications, similar to the ones by already well- studied forcing axioms such as the Proper Focing Axiom (PFA) or Martin`s Maximum (MM).However, contrary to PFA and MA, the principles RC and YPFA, do not imply Martin`s Axiom, one of the first forcing axiom ever considered. This feature is quite new and we consider therefore these two axioms are worthy of study, in order to have a better understanding of Set Theory. Hypotheses We would like to take advantage of our experience with Rado`s Conjecture to obtain further results to areas where axioms such as PFA or MM have already been successfully applied. We would like to see, for example, if the Conjecture of Rado implies every maximal almost disjoint sequence contains a Luzin sequence, among other related questions. YPFA is a weak form of PFA. We would like to verify if YPFA has however, similar consequences to PFA and Rado`s Conjecture, such as the negation of certain square principles or questions regarding properties about trees. We remark that these questions are just a starting point, but our general goal is to push forward the understanding of the properties of these two new axioms. Methods We are expecting to use traditional techniques in Set Theory such as the Pressing Down Lemma on Stationary Sets, the application of elementary submodels, walks on ordinals, etc. It is also possible that some of these desired implications do not hold. Then we will need to produce models of Rado`s Conjecture or YPFA where these implications are false. If it is the case, we need to produce models using the method of Forcing. The dissemination of results are expected to be by publications in international scientific journals and presentations on international congresses. Explanation indicating what is new and special about the project These two axioms, Rado`s Conjecture and YPFA, have the particularity that they have interesting consequences similar to traditional forcing axioms like PFA or MM, who are generalizations of Martin`s Axiom. However, neither Rado`s Conjecture nor YPFA imply Martin`s Axiom. This kind of axioms are quite new, and we consider they deserve a better understanding. A better understanding of these new axioms can have applications in other areas of Mathematics, such as Topology, Algebra or Analysis, given the well-known applications already found with PFA or MM.

We established some results between parametrized diamonds, transfinite games and invariant cardinals in a joint work with Brendle and Hrusak. We also worked with strong principles which require large cardinals such as YPFA and the P-Ideal Dichotomy. With L. Wu, we proved that the P-Ideal Dichotomy implies the negation of a two cardinal version of square principle for countable families. Also, with D. Chodounsky we proved that YPFA imply that the second uncountable cardinal has the Tree Property. Finally, with V. Di Monte, we have some consistent results regarding parametrized versions of Rado's Conjecture