If $\kappa$ is a cardinal and $X$ is a set, let $[X]^\kappa$ denote the collection of subsets of $X$ that have cardinality $\kappa$.
Let $\beta>\omega$ and $\beta \leq 2^{\omega}$. Is there ${\cal C}\subseteq [\mathbb{R}]^\beta$ such that every member of $[\mathbb{R}]^\omega$ is contained in exactly one member of ${\cal C}$?
Suppose that every countably infinite subset of $\mathbb R$ is contained in exactly one member of $\mathcal C$, where $\mathcal C\subseteq\mathcal P(\mathbb R)$ and $\mathbb R\notin\mathcal C$. Let $A$ be a countably infinite subset of $\mathbb R$. Choose a set $S\in\mathcal C$ such that $A\subseteq S$, and choose an element $t\in\mathbb R\setminus S$. Consider the countably infinite set $B=A\cup\{t\}$. Either $B$ is contained in no member of $\mathcal C$, or else $A$ is contained in two different members of $\mathcal C$; either way we have a contradiction.
If every set in $[\mathbb{R}]^{\aleph_0}$ is contained in a unique member of $\mathcal{F}$, then by induction on $\aleph_0 \leq \kappa \leq \mathfrak{c}$, it is easy to see that every set in $[\mathbb{R}]^{\kappa}$ is contained in a unique member of $\mathcal{F}$. It follows that $\mathbb{R} \in \mathcal{F}$ and hence $\mathcal{F} = \{\mathbb{R}\}$.
