# The Ramsey property implies no mad families

## TL;DR

This paper proves that the existence of the Ramsey property for all collections of infinite subsets of natural numbers implies the non-existence of infinite maximal almost disjoint families, solving a longstanding problem in set theory.

## Contribution

It establishes a deep connection between the Ramsey property and the non-existence of mad families, introducing new methods rooted in ergodic theory and descriptive set theory.

## Key findings

- If all collections of infinite subsets have the Ramsey property, then no mad families exist.
- The proof uses invariance under the $E_0$ relation to show certain functions are constant on large sets.
- Additional results relate mad families to more complex Borel ideals.

## Abstract

We show that if all collections of infinite subsets of $\N$ have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families. This solves a long-standing problem going back to Mathias \cite{mathias}. The proof exploits an idea which has its natural roots in ergodic theory, topological dynamics, and invariant descriptive set theory: We use that a certain function associated to a purported mad family is invariant under the equivalence relation $E_0$, and thus is constant on a "large" set. Furthermore we announce a number of additional results about mad families relative to more complicated Borel ideals.