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 ${\cal C}$?