Almost disjoint families and relative versions of covering properties of κ-paracompactness type

This paper is an enlarged, revised and improved version of a poster presented by the second author at the 2013 Brazilian Conference on General Topology and Set Theory (STW 2013, Maresias, Brazil, 2013). Our main goal is to investigate - within the realm of Isbell-Mrówka spaces - some relative versions of covering properties of κ-paracompactness type, inspired by a comprehensive list of strengthenings of countable paracompactness introduced by M.E. Rudin in [18]. For any property P among the ones presented, we will say that an almost disjoint family A satisfies P if it satisfies a relative version of P in the corresponding Isbell-Mrówka space. We present combinatorial characterizations of the a.d. families with some of these new relative topological properties and prove several related results; for instance, it is shown that maximal almost disjoint families are not countably paracompact. The paper finishes with a number of questions and open problems.