## Diana Piguetov\'a

*A canonical Ramsey-type theorem for finite subsets of $\Bbb N$ *

Comment.Math.Univ.Carolinae 44,2 (2003) 235-243. **Abstract:**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. \par 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.

**Keywords:** canonical coloring, forests, van der Waerden's theorem, arithmetic progression

**AMS Subject Classification:** 05C55

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