Can every infinite set be divided into pairwise disjoint subsets of size $n\in\mathbb{N}$? 
Let $S$ be an infinite set and $n$ be a natural number.  Does there exist partition of $S$ in which each subset has size $n$?



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*This is pretty easy to do for countable sets.  Is it true for uncountable sets?

*If (1) is true, can it be proved without choice?

A geometric answer, for cardinality $c$. For $x, y \in S^1$ (the unit circle), let $x \sim y$ iff $x$ and $y$ are vertices of the same regular $n$-gon centered at the origin. Now the title of this question speaks of infinite sets generally, which is a different question. The approach by Hagen von Eitzen is probably about the best you'll find in the general case.
