| Title:
|
A canonical Ramsey-type theorem for finite subsets of $\Bbb N$ (English) |
| Author:
|
Piguetová, Diana |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
44 |
| Issue:
|
2 |
| Year:
|
2003 |
| Pages:
|
235-243 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
canonical coloring |
| Keyword:
|
forests |
| Keyword:
|
van der Waerden's theorem |
| Keyword:
|
arithmetic progression |
| MSC:
|
05C55 |
| MSC:
|
05D05 |
| MSC:
|
05D10 |
| idZBL:
|
Zbl 1099.05510 |
| idMR:
|
MR2026161 |
| . |
| Date available:
|
2009-01-08T19:29:02Z |
| Last updated:
|
2020-02-20 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119383 |
| . |
| Reference:
|
[BeLe-96] Bergelson V., Leibman A.: Polynomial extension of van der Waerden's and Szemerédi's theorems.J. Amer. Math. Soc. 9 (1996), 3 725-753. MR 1325795, 10.1090/S0894-0347-96-00194-4 |
| Reference:
|
[BeLe-99] Bergelson V., Leibman A.: Set-polynomials and polynomial extension of Hales-Jewett Theorem.Ann. Math. 150 (1999), 33-75. MR 1715320, 10.2307/121097 |
| Reference:
|
[Br-00] Brown T.C.: Monochromatic forests of finite subsets of $\Bbb N$.Integers: Electronic Journal of Combinatorial Number Theory 0 (2000). MR 1759422 |
| Reference:
|
[ErGr-80] Erdös P., Graham R.L.: Old and New Problems and Results in Combinatorial Number Theory.L'Enseignement Mathématique, Genève, 1980. MR 0592420 |
| Reference:
|
[Ne-95] Nešetřil J.: Ramsey Theory.in Handbook of Combinatorics, editors R. Graham, M. Grötschel and L. Lovász, Elsevier Science B.V., 1995, pp.1333-1403. MR 1373681 |
| Reference:
|
[NeRo-84] Nešetřil J., Rödl V.: Combinatorial partitions of finite posets and lattices-Ramsey lattices.Algebra Universalis 19 (1984), 106-119. MR 0748915, 10.1007/BF01191498 |
| Reference:
|
[Ra-86] Rado R.: Note on canonical partitions.Bull. London Math. Soc. 18 (1986), 123-126. Zbl 0584.05006, MR 0818813, 10.1112/blms/18.2.123 |
| . |
