An ultrafilter is a set of subsets containing exactly one element of each finite partition: reference request There are probably dozens of ways of defining "ultrafilter".  The definition I've seen most often involves first defining "filter", then declaring an ultrafilter to be a maximal filter. 
But there's another, shorter way to state the definition:

Let $X$ be a set.  An ultrafilter on $X$ is a set $\mathcal{U}$ of subsets such that for all partitions
  $$
X = X_1 \amalg \cdots \amalg X_n
$$
  of $X$ into a finite number $n \geq 0$ of subsets, there is exactly one $i$ for which $X_i \in \mathcal{U}$.

I'd be amazed if this wasn't in the literature somewhere, but I haven't been able to track it down.  Can anyone help?
Actually, there's an even more economical definition: instead of allowing $n$ to be any natural number, you take it to be 3.  Thus, the condition is that whenever $X = A \amalg B \amalg C$, exactly one of $A$, $B$ and $C$ is in $\mathcal{U}$.  (The same thing works with 4, or 5, etc., though not with 2.)  I'm mostly interested in the version with arbitrary $n$, which seems more natural, but if you've seen the $n = 3$ version in the literature then I'd like to hear about that, too.  
Edit To be clear, when I use the word "partition" I don't mean to imply that the sets $X_i$ are nonempty.  I just mean a family of pairwise disjoint sets $X_i$ whose union is $X$.  They can be empty.
 A: The alternate formulation is closely related to the following fundamental definition from Ramsey Theory
Definition: Let $\phi : 2^X \to \lbrace\text{true},\text{false}\rbrace$ be a property pertaining to subsets of the set $X$. The property $\phi$ is called partition regular if, for every partition
$$X = X_1 \uplus X_2 \dots \uplus X_n $$ 
we have $\phi(X_i)$ for at least one $i$.
Clearly, every ultrafilter corresponds to a partition regular property, $\phi(Y) = Y\in\mathcal U$. In the other direction, it is a reasonably easy exercise to show that every partition regular property is given by a collection of ultrafilters $\phi(Y) = \bigvee \lbrace Y \in \mathcal U : \mathcal U\rbrace$. See for example theorem 3.11 in Hindman & Strauss "Algebra in the Stone-Čech compactification".

That said, I've never seen the formulation with fixed $n$, like $n=3$, before.
A: You can find that characterization (even with n = 3), as well as the generalization to $\kappa$-complete ultrafilters, in: Fred Galvin and Alfred Horn, Operations preserving all equivalence relations, Proc. Amer. Math. Soc. 24 (1970), 521-523.
A: Google's first page gave me:
Galvin  course notes Cor. 2.7...
Qiaochu Yuan blog
Maybe more on the second page?
