Maximal Almost Disjoint Families and Ramsey Regularity

Definitions and mathematical statement

Let $\omega=\{0,1,2,\ldots\}$ and let $[\omega]^\omega$ denote the set of infinite subsets of $\omega$. An infinite family $\mathcal A\subseteq[\omega]^\omega$ is almost disjoint if $a\cap b$ is finite whenever $a,b\in\mathcal A$ are distinct. Such a family is maximal almost disjoint, or mad, if it is maximal under inclusion among almost disjoint families. Equivalently, it satisfies \[ \forall b\in[\omega]^\omega\;\exists a\in\mathcal A\quad |a\cap b|=\infty. \] The requirement that $\mathcal A$ be infinite excludes finite maximal almost disjoint families. Write $\mathsf{NMAD}$ for the assertion that no infinite mad family exists.

A set $X\subseteq[\omega]^\omega$ has the Ramsey property if \[ \exists H\in[\omega]^\omega\quad \left([H]^\omega\subseteq X\ \lor\ [H]^\omega\cap X=\varnothing\right). \] Write $\mathsf{RP}$ for the assertion that every $X\subseteq[\omega]^\omega$ has this property. This gives a standard interpretation of the source's phrase "every set of reals has the Ramsey property," using infinite subsets of $\omega$ as a space of reals.

A related convention is complete Ramsey regularity. For finite $s\subseteq\omega$ and infinite $A\subseteq\omega$ with every element of $s$ less than every element of $A$, put \[ [s,A]=\{s\cup B:B\in[A]^\omega\}. \] A set $X\subseteq[\omega]^\omega$ is completely Ramsey if, for every such $s,A$, there is $H\in[A]^\omega$ such that $[s,H]\subseteq X$ or $[s,H]\cap X=\varnothing$. The supplied record does not explicitly distinguish these conventions.

Source question. Determine how $\mathsf{NMAD}$ interacts with regularity principles for sets of reals, especially universal Ramsey regularity. The record presents an open research program: it specifies neither an ambient set theory and choice assumptions, nor a complete list of regularity principles, nor a particular implication, equivalence, or relative consistency statement to establish. Consequently, it does not determine a single precise conjecture beyond this question. These missing specifications must remain unresolved in this statement-only draft.

The supplied dated assessment reports that models without mad families and models with strong regularity properties are known, and that Ramsey regularity rules out mad families. It labels the broader interaction program open; this draft records that assessment without independently verifying it or treating the reported implication as an unresolved conjecture.

Short English statement

The question concerns how the absence of infinite maximal almost disjoint families of subsets of $\omega$ relates to regularity of sets of reals, especially universal Ramsey regularity. The supplied record leaves the underlying set theory, the other regularity principles, and the exact target relationship unspecified. Its open label describes a research program requiring further formulation and human mathematical review.

Sources

• Local UnsolvedMath registry: permanent ID $30003413$, exact source problem code catalogue entry, title "Maximal Almost Disjoint Families and Ramsey Regularity," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied. The record gives proposed year 2017, status open, and statement status corrected_verified; its verification note concerns preservation of mathematical content while repairing notation, typography, and sentence structure.
• Supplied report citation: Set Theory (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/11, pp. 527--589. https://doi.org/10.4171/owr/2017/11. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions. The citation's 2018 title date and its 2017 DOI component are retained as supplied.
• Supplied literature source: On the non-existence of mad families. https://shelah.logic.at/files/94857/1090.pdf.
• Supplied literature source: The Ramsey property implies no mad families. https://arxiv.org/abs/1904.05824.
• Supplied literature source: OWR Set Theory report. https://ems.press/content/serial-article-files/46674.
• Literature assessment date: 2026-08-21. This is supplied dated curation, described in the record as literature triage derived from the OWR curation, and has not been independently verified here. It reports known models without mad families, known models with strong regularity properties, and an implication from Ramsey regularity to the absence of mad families, while labeling the broader interaction program open. The record expressly calls for independent verification of time-sensitive status and cited claims.