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Báo cáo toán học: "Exponential triple"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Exponential triples. | Exponential triples Alessandro Sisto Mathematical Institute 24-29 St Giles Oxford OX1 3LB United Kingdom sisto@maths.ox.ac.uk Submitted Mar 6 2011 Accepted Jul 6 2011 Published Jul 15 2011 Mathematics Subject Classification 05A17 Abstract Using ultrafilter techniques we show that in any partition of N into 2 cells there is one cell containing infinitely many exponential triples i.e. triples of the kind a b ab with a b 1 . Also we will show that any multiplicative IP set is an exponential IP set the analogue of an IP set with respect to exponentiation. Introduction A well-known theorem by Hindman states that given any finite partition of N there exists an infinite sets X and one cell of the partition containing the finite sums of X and also the finite products of some infinite set Y see Hi . Ultrafilters can be used to give a simpler proof than the original one see Be 1 HS . We will be interested in similar results involving exponentiation instead of addition and multiplication and our methods of proof will involve ultrafilter arguments. The first main result of this paper is the following. Theorem 1. Consider a partition of the natural numbers N A u B. Either A or B contains infinitely many triples a b ab with a b 1. Next we will provide results Theorems 14 and 15 which allow to find larger structures than the triples as above inside multiplicative IP sets see Definition 12 . A corollary of those theorems Corollary 16 is given below. Definition 2. Consider an infinite set X c N and write X x ieN with Xj Xj 1 for each j. Define inductively FEỈ 1 X yx- 1 y e FE n X u FE n X u x i FE 1 X X 1 y y e FE y X u FE y X u x i 1This is available on Bergelson s webpage http www.math.osu.edu vitaly THE ELECTRONIC JOURNAL OF COMBINATORICS v18 2011 P147 1 with FE0 X FE01 X x0 . Set FE1 X u FEn X neN FE X u F E X . neN We will say that C c N is an exponential IP set of type I resp. II if it contains a set FE1 X resp. FE11 X for some infinite X. As usual FS C and FP C denote the .