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Annals of Mathematics, II. Series Vol. 150, No. 1, pp. 3375 (1999) 

Setpolynomials and polynomial extension of the HalesJewett theoremV. Bergelson and A. LeibmanReview from Zentralblatt MATH: In this paper a generalization of the polynomial HalesJewett (PHJ) type extension of the polynomial van der Waerden theorem is proved. Its simplest variant says: If $r,d,q\in{\bbfN}$, then there exits $N(r,d,q)\in{\bbfN}$ such that, for any $r$coloring of the set of subsets of $V=\{1,\dots,N\}^d\times \{1,\dots,q\}$, there exist a set $a\subset V$, and a nonempty set $\gamma\subseteq \{1,\dots,N\}$, such that $a\cap(\gamma^d\times \{1,\dots,q\})=\emptyset$, and the sets $a$, $a\cup(\gamma^d\times \{1\})$, $a\cup(\gamma^d\times \{2\})$, \dots, $a\cup(\gamma^d\times \{q\})$ are all of the same color. The paper is divided into 9 main sections. The introduction (labeled as Section 0) is devoted to a discussion of the connections of the proved generalization of the previous related results and prepares the sole for developing the used setpolynomial machinery and the methods of topological dynamics. Here the setpolynomials are defined via the operations of union (i.e. addition) and Cartesian multiplication of finite sets. Thus setpolynomials are polynomials like expressions having finite sets as coefficients, e.g. as monomials $\gamma^d\times \{\ell\}$ of degree $d$. The dynamics is brought into play through an action $\cal{F}(W)$, where $W$ is a given set, on a topological set $X$ which is a mapping $T$ from $\cal{F}(W)$ into the set of continuous selfmappings of $X$, $a\to T^a$, satisfying that for any $a,b\in\cal{F}(W)$ with $a\cap b=\emptyset$ one has $T^{a\cup b}=T^aT^b$. PHJ can now be stated thus: Given $r,d,q\in{\bbfN}$, there exits $N(r,d,q)\in{\bbfN}$ such that, for $V=\{1,\dots,N\}^d\times \{1,\dots,q\}$ and for any $r$coloring of $\cal{F}(V)$, there exist $a\subset V$ and a nonempty set $\gamma\subseteq \{1,\dots,N\}$ such that $a\cap(\gamma^d\times \{1,\dots,q\})=\emptyset$, and the sets $a$, $a\cup(\gamma^d\times \{1\})$, $a\cup(\gamma^d\times \{2\})$, \dots, $a\cup(\gamma^d\times \{q\})$ are all of the same color. Section 1 demonstrates the method of proof of PHJ on the background of two special cases, the `linear' (i.e. $d=1$) variant of the HalesJewett theorem and the simplest nontrivial nonlinear case of a topological variant of PHJ. In Section 2 basics of setpolynomials and their formalism are developed. In Section 3 the formalism of the previous section is further developed and used to reformulate the main result in the setpolynomial language: If $A$ is a system consisting of setpolynomials with empty constant term, then $A$ is a system of (chromatic) recurrence. This result is then proved in Section 6, while Sections 4 and 5 contain some necessary technical supporting results. Section 7 is devoted to some combinatorial results derived from PHJ, and corollaries pertaining to topological recurrence. The last section is devoted to the abstract polynomiality and brings some grouptheoretical corollaries of PHJ. Reviewed by Stefan Porubský Keywords: van der Waerden theorem; geometric Ramsey theorem; setpolynomials; topological dynamics Classification (MSC2000): 05D10 05E99 11B25 37B05 54H20 Full text of the article:
Electronic fulltext finalized on: 19 Aug 2001. This page was last modified: 21 Jan 2002.
© 2001 Johns Hopkins University Press
