I've been looking into combinatorics and small cardinals, in particular, the splitting number $\mathfrak{s}$. By definition, a set $X \subseteq \omega$ splits an infinite set $Y \subseteq \omega$ if both $Y \cap X$ and $Y - X$ are inifinite.
Just out of curiosity, is splitting transitive? That is, suppose we have some infinite subset $Y$ of $\omega$, and suppose $X$ splits $Y$. From here, is it possible to split $X$ into two infinite pieces, each of of which splits $Y$?
Splitting is not transitive. Take the even numbers, $E$, and half of the odds, call them $O$. So, $O$ is a subset of the odds which also splits them. Now, $E$ splits $E\cup O$ and $E\cup O$ splits the odd numbers, but clearly $E$ doesn't split the odd numbers.
For your second question, the answer is "yes". Indeed, if $Z\subseteq X\cap Y$ is infinite, then it will split $Y$. So, given any infinite partition of $X\cap Y$ into $Z$ and $Z'$, we have that $Z, Z'$ split $X$ and $Y$.