Classically, an almost disjoint family $\mathfrak A$ on $\omega$ is a family of \emph{infinite} subsets of $\omega$ such that any two distinct elements of $\mathfrak A$ intersect finitely. A mad family is a \emph{maximal} almost disjoint family, where "maximal" means maximal under inclusion. Starting with the seminal work of Mathias in the late 1960s and beyond, and developed further by many different authors in recent years, we now have a host of theorems that tell us that we shouldn't in general expect mad families to be nicely definable (e.g. Borel, projective, etc.), unless they are finite (which is of course a slightly boring situation), and that the Ramsey property (i.e., the Baire property in the Ellentuck topology) is tightly connected to the non-definability of infinite mad families.
In this talk, I will focus on a (beautiful, to me anyhow) generalization of these non-definability results for classical mad families to the higher-dimensional analogues of mad families, where the ideal FIN of finite subsets of $\omega$ is replaced by iterated Fubini products of FIN. The main tool used to achieve these higher-dimensional results is an "operator" (or "transformation") from dimension 1 to higher dimensions, which transforms a sufficiently Ellentuck generic subset of $\omega$ (so in dimension 1) to a suitably generic set in a higher dimension; and crucially, this operator has a sort of "dimension-wise" pigeon hole property.
This is joint work with David Schrittesser.