The Ramsey property and higher dimensional mad families

TL;DR

This paper demonstrates that under a Ramsey regularity principle, infinite maximal almost disjoint families do not exist with respect to certain transfinitely iterated ideals, advancing understanding in set theory and combinatorics.

Contribution

It establishes a new non-existence result for infinite maximal almost disjoint families under Ramsey regularity assumptions.

Findings

No infinite maximal almost disjoint families exist under the given principles.

The results connect Ramsey theory with the structure of mad families.

Advances the understanding of the interplay between ideals and combinatorial properties.

Abstract

We prove that under a principle of Ramsey regularity there are no infinite maximal almost disjoint families with respect to the transfinitely iterated Fr\'echet ideals. The results of the present paper were announced by the authors in the Proceedings of the National Academy of Sciences of the U.S.A.