Ramsey Property and Pathological Sets: Almost Disjointness, Independence and Other Maximal Objects

Abstract: We show that under $\mathsf{ZF} + \mathsf{CC}<em>{\mathbb R}$, if the Ramsey property holds for all sets in a good pointclass $Γ$, then there is no MAD family in $Γ$, proving a long-standing conjecture made by A.R.D.\ Mathias in 1977. This also holds for $\mathcal I$-MAD families with respect to analytic ideals $\mathcal I$ including $\mathcal{ED}$, $\mathcal{ED}</em>{\mathrm{fin}}$, and $\finalphaα$ for all countable ordinals $α$. Under the same assumption, we show that if any one of the Baire property, Lebesgue measurability or Ramsey property holds for all sets in $Γ$, then there is no maximal independent family in $Γ$. Under the stronger assumption $\mathsf{ZF} + \mathsf{DC}_{\mathbb R}$, we further prove that if the Ramsey property holds for all sets in $Γ$, then $Γ$ contains no Vitali sets and thus no Hamel bases.

• The paper establishes that the Ramsey property precludes the existence of infinite MAD families using continuous coding and fusion techniques.
• It extends the nonexistence result to maximal independent and generalized MAD families via measure, category, and analytic ideal methods.
• The study links combinatorial pathologies with classical irregularities by showing that regularity conditions eliminate Vitali sets and Hamel bases in definable contexts.

Ramsey Property and Pathological Sets: Almost Disjointness, Independence, and Other Maximal Objects

Introduction and Context

The paper "Ramsey Property and Pathological Sets: Almost Disjointness, Independence and Other Maximal Objects" (2604.26570) investigates the relationship between classical regularity properties (notably the Ramsey property) and the existence of combinatorial and analytic pathologies in descriptive set theory. The central objects of study include maximal almost disjoint (MAD) families, maximal independent (MID) families, Vitali sets, Hamel bases, and generalizations such as $Fin^\alpha$-MAD and $\mathcal{ED}$-MAD families. The authors develop a fine-grained analysis connecting the structure of definable sets in Polish spaces to their regularity properties under various weakening of the axiom of choice.

Historically, the existence of these pathological sets (e.g., MAD families, Vitali sets) is guaranteed by classical choice principles, while models satisfying strong regularity features (such as Solovay's model with all sets measurable/Baire/Ramsey) preclude their existence. The interaction between combinatorial pathologies and hypergraph regularity, particularly via the Ramsey property, has been a central theme since Mathias' foundational work.

Main Results

Ramsey Property Implies Nonexistence of MAD Families

The primary theorem established is that under $\mathsf{ZF} + \mathsf{CC}_{\mathbb{R}}$, if every set in a sufficiently robust pointclass $\Gamma$ admits the Ramsey property, then $\Gamma$ contains no infinite MAD family. This resolves the conjecture posed by Mathias in 1977 for broad classes of definable sets, subsuming analytic, projective, and sets in Solovay's model. The proof leverages a coding operator $\Phi$ and recursive arguments to demonstrate that any candidate MAD family gives rise to a set in $\Gamma$ failing Ramsey regularity, yielding a contradiction to the global property of $\Gamma$.

The authors generalize this claim: the result extends to $\mathcal{I}$-MAD families for analytic ideals $\mathcal{I}$, including $\mathcal{ED}$0, $\mathcal{ED}$1, and $\mathcal{ED}$2 for countable $\mathcal{ED}$3. They introduce the notion of dichotomously coded ideals and establish that for such ideals, the Ramsey property precludes the existence of corresponding MAD families. The coding constructions are explicit, continuous, and adapted to the combinatorics of the underlying ideal or tree structure.

Maximal Independent Families and Classical Regularities

A separate theorem proves that within a good pointclass $\mathcal{ED}$4, if every set is Ramsey, Lebesgue measurable, or has the Baire property, then no MID family exists in $\mathcal{ED}$5. The proof decomposes $\mathcal{ED}$6 into countable unions of filters and ideals and demonstrates, via classical measure and category results, that this would force the entire Cantor space to be null or meager, contradicting fundamental theorems (measure-theoretic or category-theoretic).

Notably, while Lebesgue measurability may permit MAD families in certain models, it never allows MID families, indicating that MID families are at least as sensitive to definable regularity as MAD families.

Ramsey Property and Nonexistence of Vitali Sets and Hamel Bases

Under $\mathcal{ED}$7, if every set in $\mathcal{ED}$8 has the Ramsey property, then $\mathcal{ED}$9 contains no Vitali sets and, consequently, no Hamel bases. The argument is based on the logic that a Hamel basis yields a Vitali set within the same pointclass by algebraic representation, and the existence of a continuous coding reduces non-Ramsey behavior to one-dimensional pathologies. This result extends the impact of Ramsey regularity beyond purely combinatorial objects, influencing classical analytic irregularities.

Extending to Generalized MAD Families

For the recently studied generalized MAD families ($\mathsf{ZF} + \mathsf{CC}_{\mathbb{R}}$0-MAD for countable $\mathsf{ZF} + \mathsf{CC}_{\mathbb{R}}$1, $\mathsf{ZF} + \mathsf{CC}_{\mathbb{R}}$2-MAD, and $\mathsf{ZF} + \mathsf{CC}_{\mathbb{R}}$3-MAD), the paper provides explicit coding and splitting constructions establishing their nonexistence under the Ramsey property for sets in a good pointclass. No additional technical assumptions (such as the strengthening $\mathsf{ZF} + \mathsf{CC}_{\mathbb{R}}$4-Unif) are required, and the proofs are self-contained and direct.

Methodological Features and Technical Claims

The main technical innovation is the construction of continuous coding operators and recursive fusion arguments exploiting the Ramsey property at all definable levels. These operators map infinite sequences to combinatorial configurations (subsets, branches, etc.) and encode interactions with maximal families. The proof strategies generalize well-known fusion methods in infinite combinatorics to the descriptive set-theoretic context.

The paper emphasizes the dichotomous splitting property at the heart of these constructions: for any infinite set, the coding yields configurations that force non-Ramsey (non-homogeneous) behavior, showing that maximality and almost disjointness are fundamentally incompatible with global regularity.

For filters and ideals, classical results (Sierpiński, Talagrand, Mathias) are deployed to demonstrate that the regularity properties annihilate the possibility of non-meager/measurable filters/ideals, precluding MID structures.

Strong claims include:

• The Ramsey property for all sets in a robust pointclass (e.g., analytic, projective, or Solovay's model) is sufficient to rule out the existence of infinite MAD families in that pointclass.
• This holds not only for classical MAD families but also for generalizations (including those associated with analytic ideals and transfinitely iterated Frechet ideals), without requiring additional uniformization principles.
• Any definable maximal independent family is destroyed by regularity properties, including measurability, Baire, or Ramsey.
• Application to analytic pathologies: No analytic Vitali set, Hamel basis, MAD, MID, or generalized MAD families exist if all analytic sets satisfy Ramsey regularity.

Implications and Further Directions

The results provide a unified approach to the destruction of combinatorial and analytic pathologies under strong regularity, encompassing classical objects (Vitali sets, Hamel bases) and modern generalizations (various MAD/MID families). Practically, this refines our understanding of the landscape of definable sets in Polish spaces under weak choice principles and determinacy assumptions.

Theoretically, the extension to dichotomously coded ideals and the explicit construction of continuous coding operators suggest new avenues for classifying which ideals and maximal configurations are, in principle, resilient to regularity versus those inherently destroyed.

Open questions and future directions include:

• The consistency strength and independence of infinite MAD families under weakened forms of choice and regularity.
• Classification of analytic ideals for which no analytic MAD family exists.
• Existence and structure of definable MAD/MID families in generic extensions and their relationship with dominating and unbounded reals.
• The role of coding operators and fusion constructions in more general hypergraph independence settings.

The paper prompts further investigation into the minimal definable complexity of idealized MAD families, the nature of forcing indestructibility, and the interplay between large cardinal assumptions, determinacy, and combinatorial regularity.

Conclusion

This work rigorously delineates the role of the Ramsey property and related regularity conditions in excluding pathological combinatorial and analytic objects from robust pointclasses. By combining coding and fusion arguments, measure-category results, and explicit combinatorial constructions, the authors resolve longstanding conjectures, dramatically expand the scope of known results, and lay foundations for further exploration of the boundary between definability, pathology, and regularity in descriptive set theory (2604.26570).