Maximal Ramsey families

Question

We say that a family $\mathcal R\subseteq \mathcal P(\omega)$ is Ramsey if

1. $\bigcup \mathcal R = \omega$, and
2. for every map $f:\mathcal R \to \{0,1\}$ there is an infinite set $X\subseteq \omega$ such that the restriction $f|_{\mathcal P(X)\cap \mathcal R}:\bigl(\mathcal P(X)\cap \mathcal R\bigr) \to \{0,1\} $ is constant.

Question. Is every Ramsey family contained in a Ramsey family that is maximal with respect to $\subseteq$?

Answer 1

No. Consider the family $\mathcal{R}$ of all sets of size 1. It is certainly a Ramsey family. Consider a Ramsey extension $\mathcal{R}_1\supset \mathcal{R}$. Take a map $f$ which maps all sets from $\mathcal{R}$ to 0 and all sets from $\mathcal{R}_1\setminus \mathcal{R}$ to 1. On a monochromatic infinite $X$, the constant value of $f$ is of course 0. Thus, this $X$ does not contain a set from $\mathcal{R}_1\setminus \mathcal{R}$. Take every subset $A\subset X$ of size 2 and add it to $\mathcal{R}_1$. We still get a Ramsey family, as is seen from restricting everything to $X\setminus A$. Thus $\mathcal{R}_1$ is not an inclusion-maximal extension.