Set-polynomials and polynomial extension of the Hales-Jewett Theorem

• 成果类型: Article
• 署名作者: Bergelson, V; Leibman, A
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/121097
• 发表日期: 1999
• 页码: 33-75
• 关键词: ramsey theory

摘要

An abstract, Hales-Jewett type extension of the polynomial van der Waerden Theorem [BL] is established: THEOREM. Let r,d,q is an element of N. There exists N is an element of N such that for any r-coloring of the set of subsets of V = (1,..., N)(d) x (1,...,q) there exist a set a subset of V and a nonempty set gamma subset of or equal to (1,..., N) such that a boolean AND (gamma(d) x (1,..., q)) = 0 and the subsets a, a boolean OR (gamma(d) x (1)), a boolean OR (gamma(d) x (2)),..., a boolean OR (gamma(d) x (q)) are all of the same color. This polynomial Hales-Jewett theorem contains refinements of many combinatorial facts as special cases. The proof is achieved by introducing and developing the apparatus of set-polynomials (polynomials whose coefficients are finite sets) and applying the methods of topological dynamics.