How many different colors do we need so that the set of all possible colorings of $\mathbb{R}^3$ is greater than the powerset of $\mathbb{R}$. Countably many doesn't seem to be enough and even $|\mathbb{R}|$ seems insufficient. I asked this on stackexchange but there is no answer. Is the powerset of $\mathbb{R}$ the least cardinality of the set of colors for this to hold?