Let $\kappa$ be an infinite cardinal and as usual denote by $[\kappa]^\mu$ the set of all subsets of $\kappa$ having cardinality $\mu$:

Call a family $\mathcal{F} \subset [\kappa]^\omega$

sparseif for all $\mathcal{G} \in [\mathcal{F}]^{\omega_1}$ the set $\bigcup \mathcal{G}$ is uncountable.

This is a pretty natural notion that has been rediscovered by various authors and is the combinatorial core of various seemingly unrelated problems in topology, analysis and algebra (see the list of references at the end).

Getting a sparse family isn't that much of a deal. The problem is often getting a big one, or more precisely, a *cofinal* one, with respect to containment.

It's easy to see that there is a sparse cofinal family in $([\aleph_n]^\omega, \subseteq)$), for every $n<\omega$. This is a simple consequence of the fact that $cf([\aleph_n]^\omega, \subseteq)=\aleph_n$.

QUESTION: Is there (in ZFC) a cardinal $\kappa$ such that $cf([\kappa]^\omega, \subseteq) > \kappa$ and there is a sparse cofinal family on $[\kappa]^\omega$?

This cardinal, if it exists, must be larger than $\aleph_\omega$, because the existence of a sparse cofinal family of countable subsets of $\aleph_\omega$ can be proved to be independent of ZFC, modulo very large cardinals. Indeed (see Todorcevic's book or Blass's article for the proofs):

If the Chang's Conjecture variant $(\aleph_{\omega+1}, \aleph_\omega) \twoheadrightarrow (\aleph_1, \aleph_0)$ holds then there is no sparse cofinal family of countable subsets of $\aleph_\omega$. The consistency of this Chang's Conjecture variant has been proven from (slightly less than) a 2-huge cardinal by Levinski, Magidor and Shelah.

If $\square_{\aleph_\omega}+cf([\aleph_\omega]^\omega, \subseteq)=\aleph_{\omega+1}$ holds then there is a sparse cofinal family of subsets of $\aleph_\omega$.

REFERENCES:

*Blass, Andreas*,**On the divisible parts of quotient groups**, Göbel, Rüdiger (ed.) et al., Abelian group theory and related topics. Conference, August 1-7, 1993, Oberwolfach, Germany. Providence, RI: American Mathematical Society. Contemp. Math. 171, 37-50 (1994). ZBL0823.20057.*Kojman, Menachem; Milovich, David; Spadaro, Santi*,**Noetherian type in topological products**, Isr. J. Math. 202, 195-225 (2014). ZBL1302.54008.Peter Nyikos,

*Generalized Kurepa and MAD families and Topology*, preprint.*Spadaro, Santi*,**On two topological cardinal invariants of an order-theoretic flavour**, Ann. Pure Appl. Logic 163, No. 12, 1865-1871 (2012). ZBL1269.03047.*Todorcevic, Stevo*, Walks on ordinals and their characteristics, Progress in Mathematics 263. Basel: Birkhäuser (ISBN 978-3-7643-8528-6/hbk). vi, 324 p. (2007). ZBL1148.03004.

high Davies trees, which are defined in the Soukups' paper arxiv.org/abs/1705.06195, and whose existence follows from $\mathsf{GCH}+\square$. $\endgroup$