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.
@article{119383,
author = {Diana Piguetov\'a},
title = {A canonical Ramsey-type theorem for finite subsets of $\Bbb N$},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {44},
year = {2003},
pages = {235-243},
zbl = {1099.05510},
mrnumber = {2026161},
language = {en},
url = {http://dml.mathdoc.fr/item/119383}
}
Piguetová, Diana. A canonical Ramsey-type theorem for finite subsets of $\Bbb N$. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) pp. 235-243. http://gdmltest.u-ga.fr/item/119383/
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