Logic Seminar

Many are familiar with the classic result due to Mathias that there are no analytic infinite maximal almost disjoint families, where almost disjoint means "having finite intersection". In this talk I will discuss what happens if we replace "having finite intersection" with "having intersection in a (fixed) ideal I", where I is a Borel ideal on $\omega$. It turns out that for a large class of ideals it is possible to prove an analogue of Mathias' theorem, while for other ideals there _are_ definable infinite mad families. No characterization of when one or the other situation arises seems to be known at this time.