Geodesic
A canonical Ramsey-type theorem for finite subsets of $\Bbb N$
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 2, pp. 235-243.

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T. Brown proved that whenever we color $\Cal P_{f} (\Bbb N)$ (the set of finite subsets of natural numbers) with finitely many colors, we find a monochromatic structure, called an arithmetic copy of an $\omega $-forest. In this paper we show a canonical extension of this theorem; i.e\. whenever we color $\Cal P_{f}(\Bbb N)$ with arbitrarily many colors, we find a canonically colored arithmetic copy of an $\omega $-forest. The five types of the canonical coloring are determined. This solves a problem of T. Brown.
Classification : 05C55, 05D05, 05D10
Keywords: canonical coloring; forests; van der Waerden's theorem; arithmetic progression 
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     title = {A canonical {Ramsey-type} theorem for finite subsets of $\Bbb N$},
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Piguetová, Diana. A canonical Ramsey-type theorem for finite subsets of $\Bbb N$. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 2, pp. 235-243. http://geodesic.mathdoc.fr/item/CMUC_2003__44_2_a4/