Ramsey regularity implies no MAD families without uniformization

TL;DR

This paper proves that the Ramsey property in a class of sets prevents the existence of MAD families under certain conditions, confirming a conjecture from 1977, and introduces a versatile technique for related problems.

Contribution

It establishes a link between Ramsey properties and the non-existence of MAD families, and introduces a new proof technique applicable to various problems.

Findings

No MAD families exist under Ramsey property assumptions.

Introduces a novel proof technique for related combinatorial problems.

Announces four related theorems to be proved in future work.

Abstract

We show that if the Ramsey property holds (in a class of sets), then there is no MAD family (in this class, provided it satisfies some modest closure properties), proving a conjecture made by A.R.D.\ Mathias in 1977. As the technique we introduce for this proof is useful in a variety of related problems, we take the opportunity to announce 4 theorems, which will be proved in a follow-up paper.