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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?

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According to P. Erdõs, A. Hajnal: On a property of families of sets, Acta Math. Acad. Sci. Hungar. 12 (1961), 87--123 (see page 90) the stronger result you formulated is a theorem of Bernstein from F. Bernstein, "Zur Theorie der trigonometrischen Reihen", Leipz. Bet. (Berichte über die Verhandlungen der Königl. Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Math.phys. Klasse) 60 (1908), 325--338" MR, eudml. 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 may contain some results you are interested in.

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    $\begingroup$ I conjecture that the part in parentheses (beginning "Beriehte") should be "Berichte über die Varhandlungen der Königl. Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Math.phys. Klasse". $\endgroup$ – Andreas Blass Nov 1 '15 at 17:43
  • $\begingroup$ Thank you, Andreas. The copy-paste from a pdf file changed some letter. $\endgroup$ – Lajos Soukup Nov 1 '15 at 18:52
  • $\begingroup$ I have made the change suggested by Andreas. $\endgroup$ – LSpice Nov 1 '15 at 19:20

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