Let $S$ is a partition of a set $U$. Let $c$ is an ultrafilter on $U$.


Prove or disprove this conjecture:


At least one of the following is true:


* $\exists D\in S, C\in c:C\subseteq D$
or
* $\exists C\in c\forall D\in S: \mathrm{card}(C\cap D)\le 1$.



An ultrafilter and a partition
Take a partition of ${\mathbb N}^2$ into vertical lines $\{x\}\times {\mathbb N}$. In each vertical line take a non-principle ultrafilter $\omega\_x$. Now take the set of all sets $Y$ that intersect all but finitely many vertical lines by a subset from $\omega\_x$. Note that all complements of finite sets of ${\mathbb N}^2$ are in our set of sets, and that it is clearly a filter. Take any ultrafilter $\omega$ that contains that filter. It exists by the Zorn lemma. Clearly, the first option does not hold: none of the vertical lines is in $\omega$. Now suppose that for some $C\in\omega$, $C$ intersects each vertical line by at most 1 element. Then its complement intersects each vertical line by a subset that is either the whole line or the line without one element. That is impossible because we chose non-principle $\omega\_x$. Thus $\omega$ is a counterexample. 


