Singular cardinals and cardinal characteristics

This project deals with the study of cardinal characteristics of the generalized Baire spaces , in the specific case where is an uncountable singular cardinal. It is inspired by the work of Shimon Garti and Saharon Shelah on the study of cardinal characteristics of generalized Baire spaces, for an uncountable cardinal. Among others, we will study cardinals in the generalized Cichon`s diagram and the consequences the singularity assumption will have on it, by using the already existing tools in the area of singular cardinals and our experience in the study of such generalizations. Cardinal characteristics of the classical Baire space are cardinals describing mostly the combinatorial or topological structure of the real line. They are usually defined in terms of ideals on the reals, or some very closely related structure such as ()/ and typically they assume values between 1 , the first uncountable cardinal and . Hence, they are uninteresting in models where the continuum hypothesis (20 = 1 ) holds. However, in models of set theory where CH fails they may assume different values and interact with each other in several ways. In the last years, special interest has been given to the study of these characteristics on the generalized Baire spaces (the space of functions from to ), when is an uncountable cardinal. By the time, the case where is additionally regular (or even a large cardinal) has been explored by many authors (included me) and nowadays it is possible to find many interesting ZFC and consistency results involving them. On the other hand, singular cardinals represent another important area of study within set theory. They arose from the crucial concept of cofinality, which appeared after Julius König. These cardinals turn to be the source of many interesting problems and were the starting point of the outstanding PCF theory (PCF stands for possible cofinalities.) Particular interest has been given, for instance to the value of the continuum function for singular cardinals which have risen to the well- know singular cardinal hypothesis (SCH).

The main results obtained during the two years of this project are related to the study of maximal almost disjoint families and maximal independent families of the generalized Baire spaces $\lambda^\lambda$ when $\lambda$ is a singular cardinal. In regards to the almost disjointness number for singular cardinals and based on the results of Erdös and Shelah and Kojman, Kubi\'s and Shelah, we built a model in which the inequality $\fra(\lambda) < \fra$ for $\lambda$ a singular cardinal of countable cofinality holds. We also proved some results dealing with the concept of destroying the maximality of a given maximal almost disjoint family at a singular cardinal $\lambda$ by using forcing. Moreover, we proved a preservation result of madness when changing the cofinality of a given large cardinal $\kappa$. Additionally, related to maximal independence, we studied the concept of independence of families of subsets of a singular cardinal $\lambda$. First, we proved several basic results and compare them with the properties of independent families on regular cardinals. The main results of these investigations shows that assuming the existence of large cardinals, it is possible to construct a maximal independent family at a singular cardinal $\lambda$ which is a strong limit for large cardinals.