A canonical Ramsey-type theorem for finite subsets of

Diana Piguetová

Commentationes Mathematicae Universitatis Carolinae (2003)

  • Volume: 44, Issue: 2, page 235-243
  • ISSN: 0010-2628

Abstract

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T. Brown proved that whenever we color 𝒫 f ( ) (the set of finite subsets of natural numbers) with finitely many colors, we find a monochromatic structure, called an arithmetic copy of an ω -forest. In this paper we show a canonical extension of this theorem; i.eẇhenever we color 𝒫 f ( ) with arbitrarily many colors, we find a canonically colored arithmetic copy of an ω -forest. The five types of the canonical coloring are determined. This solves a problem of T. Brown.

How to cite

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Piguetová, Diana. "A canonical Ramsey-type theorem for finite subsets of $\mathbb {N}$." Commentationes Mathematicae Universitatis Carolinae 44.2 (2003): 235-243. <http://eudml.org/doc/249205>.

@article{Piguetová2003,
abstract = {T. Brown proved that whenever we color $\mathcal \{P\}_\{f\} (\mathbb \{N\})$ (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.eẇhenever we color $\mathcal \{P\}_\{f\}(\mathbb \{N\})$ 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.},
author = {Piguetová, Diana},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {canonical coloring; forests; van der Waerden's theorem; arithmetic progression; canonical coloring; forests; van der Waerden's theorem; arithmetic progression},
language = {eng},
number = {2},
pages = {235-243},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A canonical Ramsey-type theorem for finite subsets of $\mathbb \{N\}$},
url = {http://eudml.org/doc/249205},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Piguetová, Diana
TI - A canonical Ramsey-type theorem for finite subsets of $\mathbb {N}$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 2
SP - 235
EP - 243
AB - T. Brown proved that whenever we color $\mathcal {P}_{f} (\mathbb {N})$ (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.eẇhenever we color $\mathcal {P}_{f}(\mathbb {N})$ 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.
LA - eng
KW - canonical coloring; forests; van der Waerden's theorem; arithmetic progression; canonical coloring; forests; van der Waerden's theorem; arithmetic progression
UR - http://eudml.org/doc/249205
ER -

References

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  1. Bergelson V., Leibman A., Polynomial extension of van der Waerden's and Szemerédi's theorems, J. Amer. Math. Soc. 9 (1996), 3 725-753. (1996) MR1325795
  2. Bergelson V., Leibman A., Set-polynomials and polynomial extension of Hales-Jewett Theorem, Ann. Math. 150 (1999), 33-75. (1999) MR1715320
  3. Brown T.C., Monochromatic forests of finite subsets of , Integers: Electronic Journal of Combinatorial Number Theory 0 (2000). (2000) MR1759422
  4. Erdös P., Graham R.L., Old and New Problems and Results in Combinatorial Number Theory, L'Enseignement Mathématique, Genève, 1980. MR0592420
  5. Nešetřil J., Ramsey Theory, in Handbook of Combinatorics, editors R. Graham, M. Grötschel and L. Lovász, Elsevier Science B.V., 1995, pp.1333-1403. MR1373681
  6. Nešetřil J., Rödl V., Combinatorial partitions of finite posets and lattices-Ramsey lattices, Algebra Universalis 19 (1984), 106-119. (1984) MR0748915
  7. Rado R., Note on canonical partitions, Bull. London Math. Soc. 18 (1986), 123-126. (1986) Zbl0584.05006MR0818813

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