Almost-Disjointness Numbers at Singular Cardinals

Definitions and mathematical statement

For an infinite cardinal $\kappa$, write $[\kappa]^\kappa=\{A\subseteq\kappa:|A|=\kappa\}$. A family $\mathcal A\subseteq[\kappa]^\kappa$ is almost disjoint if $|A\cap B|<\kappa$ for distinct $A,B\in\mathcal A$. It is maximal almost disjoint if no additional member of $[\kappa]^\kappa$ can be adjoined while preserving this property; equivalently, every $B\in[\kappa]^\kappa$ has $|A\cap B|=\kappa$ for some $A\in\mathcal A$.

Interpreting the source's cofinality lower bound as a restriction on the families over which the minimum is taken, define \[ \mathfrak a_\kappa=\min\{|\mathcal A|: \mathcal A\subseteq[\kappa]^\kappa\text{ is maximal almost disjoint},\quad |\mathcal A|\geq\operatorname{cf}(\kappa)\}, \qquad \mathfrak a=\mathfrak a_\omega. \] This restriction excludes the trivial small maximal families; the intended convention at singular cardinals should be confirmed from the report.

The two separate consistency questions are whether the respective theories \[ \mathrm{ZFC}+\left(\mathfrak a_{\aleph_\omega}<\mathfrak a \ \text{and}\ \mathfrak a>\aleph_\omega\right) \] and \[ \mathrm{ZFC}+\left(\mathfrak a_{\aleph_{\omega_1}}<\mathfrak a_{\aleph_1}\right) \] are consistent. Here $\omega_1$ is the first uncountable ordinal, so the cardinal index in the second question is $\aleph_{\omega_1}$. The extra lower bound on $\mathfrak a$ is explicitly part of the first question only. No simultaneous consistency claim, specified forcing construction, or large-cardinal hypothesis is supplied.

Short English statement

Can the almost-disjointness number at $\aleph_\omega$ be below the classical $\mathfrak a$ while $\mathfrak a>\aleph_\omega$? Separately, can $\mathfrak a_{\aleph_{\omega_1}}$ be below $\mathfrak a_{\aleph_1}$?

Sources

• Local UnsolvedMath registry, permanent ID $30000904$, source code catalogue entry, "Almost-Disjointness Numbers at Singular Cardinals." No individual proposer is supplied.
• Set Theory (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/02, pp. 79--118: https://doi.org/10.4171/owr/2008/02.
• Supplied reference, "singular madness results": https://shelah.logic.at/files/182061/793.pdf.
• Supplied literature triage, checked 2026-08-21: status "open." It says the report poses both inequalities as open and that a known consistency result does not decide the requested conditions, mentioning $\mathfrak a>\aleph_\omega$. That result and its hypotheses are not reproduced. This dated assessment derives from supplied OWR curation; time-sensitive status and cited claims require independent verification.