Maximal almost disjoint families—i.e., collections of infinite sets of integers with pairwise finite intersection, which are maximal with respect to inclusion—have been a ubiquitous object of study in set theory for the past half century. Aside from their applications in topology and functional analysis, one can ask about the possible uncountable cardinals for which there exists a mad family of that size, and one can also wonder as to the potential definability of mad families, in the sense of classical descriptive set theory. Each of these questions has and continues to motivate two rich body of research in set theory of the reals and descriptive set theory, respectively; see, for example, [8], [11], [3], [5], and [9], [12], [10].
More recently, authors such as Brendle, Fischer, S.D. Friedman, Khomskii, and Zdomskyy have been interested in combining these two interests, namely by asking how to construct models of set theory in which we control the realization of the set
sp(a) = { |A| : A is a mad family},
while also ensuring that for some κ ∈ sp(a), there is a projectively definable mad family with definition of minimal descriptive complexity. In this talk I will briefly overview some known results connecting definability, cardinality, and almost disjointness, discussing the obstacles one may encounter and strategies to overcome them. In particular, I will delve into a construction from my thesis, giving a model in which the mad spectrum contains only two cardinals, each of whom is witnessed by a projective mad family with an additional strong combinatorial property.
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[5] Vera Fischer and Julia Millhouse Strong projective witnesses. Submitted, 2025.
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