The alternate formulation is closely related to the following fundamental definition from [Ramsey Theory][1]

**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][2] in Hindman & Strauss "Algebra in the Stone-Čech compactification".

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That said, I've never seen the formulation with fixed $n$, like $n=3$, before.


   [1]: http://en.wikipedia.org/wiki/Ramsey_theory
   [2]: http://books.google.com/books?id=KYXgdiegKDsC&pg=PA51&lpg=PA51&dq=hindman+strauss+ultrafilter+3.11&source=bl&ots=AxkxqyCuxv&sig=40KaioHGEw2wRJtHI6V3_0E6s4I&hl=en&ei=2sQRTsS8CcntOey-2bkL&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBQQ6AEwAA#v=onepage&q&f=false