The Ramsey property, MAD families, and their higher dimensional relatives

21.06.2022 15:00 - 16:30

D. Schrittesser (U of Toronto, CA)

Infinite maximal almost disjoint families, dubbed "MAD families" by A.R.D. Mathias, have long been an object of interest in set theory, topology, and other areas. A question which has been tossed around for quite a while was whether there can exist an analytic Fin2-MAD family - that is, a two-dimensional variant of the usual notion of MAD family. Analytic (one-dimensional) MAD families cannot exist, so the conjecture has always been "no" - and indeed the answer is "no", as was shown in joint work with Törnquist and Bakke Haga in 2016.

In 1969, Mathias asked whether Ramsey regularity rules out the existence of (one dimensional) MAD families. This question was answered positively in joint work with Törnquist in 2019.

But now that we know that Fin2 MAD families behave like MAD families in some respects, and that Ramsey regularity rules out (one dimensional) MAD families, does Ramsey regularity also rule out the two dimensional variant?

Yes, Ramsey regularity rules out the existence of Fin2 MAD families (also joint work with Asger Törnquist). The result even holds for J-MAD families, where J is an ideal in the smallest class containing the ideal of finite sets and closed under Fubini products. I will also report on work in progress with our student Severin Mejak.




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