$\newcommand{\H}{\mathcal{H}}$
$\newcommand{\A}{\mathcal{A}}$
Recall that a coideal $\H$ over $\omega$ is **selective** if for every $\{A\_n : n < \omega\} \subseteq \H$, where $i < j \implies A\_i \supseteq A\_j$, there exists some $B \in \H$ such that $B/n := \{m \in B : m > n\} \subseteq A\_n$ for all $n \in B$. There are two canonical examples of selective coideals, both of which can be constructed in $\mathsf{ZFC}$:


* The entire $[\omega]^\omega$ is of course a selective coideal.
* Let $\A$ be a mad (maximal almost disjoint) family of subsets of $\omega$. Let $\H$ be the set of all infinite subsets of $\omega$ which cannot be covered up to a finite set by finitely many members of $\A$. Then $\H$ is a selective coideal.


Are there any more examples of selective coideals that $\mathsf{ZFC}$ can prove to exist? Note that I'm not looking for selective coideals that exist with further assumptions (e.g. under $\mathsf{CH}$, there are $2^{2^{\aleph\_0}}$ many Ramsey ultrafilters, but I'm not asking for such examples).



Non-trivial examples of selective coideals of $\omega$
See Section 12 in [S. Todorcevic, Topics in Topology, Lecture notes in Mathematics Vol. 1652, Springer-Verlag Berlin Heidelberg 1997](https://doi.org/10.1007/BFb0096295)


