# A kind of anti-Ramsey result

In contrast to classic results for arithmetic progressions of arbitrary length in one set at least of any finite partition of $\mathbb N$, it is easy to construct a partition in two sets of integers $A$ and $B$ such that neither $A$ nor $B$ contains an infinite arithmetic sequence. Actually (with choice, of course), one has a much stronger result : if $(A_i)_{i\in I}$ is a family of subsets of $E$ with $\#A_i=\#I=\#E$ (infinite), there exists a partition of $E$, $E=F\cup G$, such that for all $i\in I$, $A_i$ is not included in $F$ nor in $G$ (the not very hard proof using a diagonal argument must be well-known). Is this kind of anti-Ramsey result part of some more general theory?