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?



A kind of anti-Ramsey result
According to [P. Erdõs, A. Hajnal: On a property of families of sets, Acta Math. Acad. Sci. Hungar. 12 (1961), 87--123](http://www.renyi.hu/~p_erdos/1961-11.pdf) (see page 90) the stronger result you formulated is a theorem of Bernstein from *F. Bernstein, "[Zur Theorie der trigonometrischen Reihen](https://doi.org/10.1515/crll.1907.132.270)", Leipz. Bet. (Berichte über die Verhandlungen der Königl. Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Math.phys. Klasse) 60 (1908), 325--338*" [MR](http://www.ams.org/mathscinet-getitem?mr=1580725), [eudml](https://eudml.org/doc/149267). If you are looking for further results, "property B" is the keyword you need. 


I presented a minicourse on some related properties of families of sets on the 7th Young Set Theory Workshop in 2014, so [my slides](http://bcc.impan.pl/14Young/index.php/slides) may contain some results you are interested in.


