# Article

Full entry | PDF   (0.2 MB)
Keywords:
canonical coloring; forests; van der Waerden's theorem; arithmetic progression
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.
References:
[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
[BeLe-99] Bergelson V., Leibman A.: Set-polynomials and polynomial extension of Hales-Jewett Theorem. Ann. Math. 150 (1999), 33-75. MR 1715320
[Br-00] Brown T.C.: Monochromatic forests of finite subsets of $\Bbb N$. Integers: Electronic Journal of Combinatorial Number Theory 0 (2000). MR 1759422
[ErGr-80] Erdös P., Graham R.L.: Old and New Problems and Results in Combinatorial Number Theory. L'Enseignement Mathématique, Genève, 1980. MR 0592420
[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
[NeRo-84] Nešetřil J., Rödl V.: Combinatorial partitions of finite posets and lattices-Ramsey lattices. Algebra Universalis 19 (1984), 106-119. MR 0748915
[Ra-86] Rado R.: Note on canonical partitions. Bull. London Math. Soc. 18 (1986), 123-126. MR 0818813 | Zbl 0584.05006

Partner of