Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Two New Extensions of the Hales-Jewett Theorem. | Two New Extensions of the Hales-Jewett Theorem Randall McCutcheon Department of Mathematics University of Maryland College Park MD 20742 randall@ Submitted June 30 2000 Accepted September 28 2000 Abstract We prove two extensions of the Hales-Jewett coloring theorem. The first is a polynomial version of a finitary case of Furstenberg and Katznelson s multiparameter elaboration of a theorem due to Carlson about variable words. The second is an idempotent version of a result of Carlson and Simpson. MSC2000 Primary 05D10 Secondary 22A15. For k N 2 N let W N denote the set of length-N words on the alphabet 0 1 k 1 . A variable word over WN is a word w x of length N on the alphabet 0 1 k 1 x in which the letter x appears at least once. If w x is a variable word and i 2 0 1 . k 1 we denote by w i the word that is obtained by replacing each occurrence of x in w x by an i. The Hales-Jewett theorem states that for every k r 2 N there exists N N k r 2 N such that for any partition Wk Ji 1 Ci there exist j 1 j r and a variable word w x over WN such that w i i 2 0 . k 1 c Cj. l. Finitary extensions. In BL V. Bergelson and A. Leibman provided a polynomial version of the Hales-Jewett theorem. In order to formulate their result we must develop some terminology. Let l 2 N. A set-monomial over Nl in the variable X is an expression m X S1 X S2 X X Si where for each i 1 i l Si is either the symbol X or a nonempty singleton subset of N these are called coordinate coefficients . The degree of the monomial is the number of times the symbol X appears in the list S1 Si. For example taking l 3 m X 5 X X X X is a set-monomial of degree 2 while m X X X 17 X 2 is a set-monomial of degree 1. A set-polynomial is an expression of the form P X m1 X u m2 X u u Ilk X where k 2 N and m1 X mk X are set-monomials. The degree of a set-polynomial is the largest degree of its set-monomial summands and its constant term consists of the sum of The author acknowledges support from the National .
