Ramsey regularity implies no MAD families without uniformization
Jialiang He, Jintao Luo, Shuguo Zhang
TL;DR
The paper addresses Mathias's conjecture that Ramsey regularity excludes certain irregular sets by proving a general result: if a good pointclass $\Gamma$ consists entirely of Ramsey-regular sets, then there are no Dedekind infinite MAD families in $\Gamma$. The authors work in $\mathrm{ZF}$ and assume a hypothetical Dedekind infinite MAD family $\mathcal{A}\subseteq\Gamma$, reducing to a disjoint sequence $A_n$ and introducing the transform $\widetilde{z}$ along with the derived set $P=\{z: \exists A\in\mathcal{A}\, \widetilde{z}\subseteq A\}$; since $\Gamma$ is good, $P\in\Gamma$, and two key Claims yield a contradiction to the Ramsey regularity of $P$. This yields the main theorem: no Dedekind infinite MAD family exists in $\Gamma$ whenever every set in $\Gamma$ has the Ramsey property. The paper also announces several follow-up results showing broader nonexistence consequences (e.g., Vitali sets, Hamel bases, maximal independent families, and various $\mathcal{I}$-MAD variants) within the same framework, underlining the robustness of the method beyond the original conjecture.
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.