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canonical coloring; forests; van der Waerden's theorem; arithmetic progression
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.
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