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: On a Rado Type Problem for Homogeneous Second Order Linear Recurrences. | On a Rado Type Problem for Homogeneous Second Order Linear Recurrences Hayri Ardal Zdenek Dvorak Veselin Jungic Tomas Kaiser Submitted Sep 24 2009 Accepted Feb 23 2010 Published Mar 8 2010 Mathematics Subject Classification 05D10 Abstract In this paper we introduce a Ramsey type function S r a b c as the maximum s such that for any r-coloring of N there is a monochromatic sequence x1 x2 . xs satisfying a homogeneous second order linear recurrence aXi bxi 1 cxi 2 0 1 i s 2. We investigate S 2 a b c and evaluate its values for a wide class of triples a b c . 1 Introduction In this paper we are interested in the following question If the set of positive integers N is finitely colored is it possible to find a monochromatic sequence of a certain length that satisfies a given second order homogeneous recurrence A reader that is even remotely familiar with Ramsey Theory would quickly note that Van der Waerden s theorem affirmatively answers this question for the recurrence xi 2xi 1 xi 2 0 any finite coloring of N and any finite sequence length. But what about other second order homogeneous recurrences In 1997 Harborth and Maasberg 4 considered the recurrence xi xi 1 axi 2 and obtained a puzzling sequence of results that have inspired a large portion of the work presented in this paper Department of Mathematics Simon Fraser University Burnaby . V5A 2R6 Canada. E-mail hardal@. Department of Mathematics Simon Fraser University Burnaby . V5A 2R6 Canada. E-mail rakdver@. Department of Mathematics Simon Fraser University Burnaby . V5A 2R6 Canada. E-mail vjungic@. Department of Mathematics and Institute for Theoretical Computer Science University of West Bohemia Univerzitni 8 306 14 Plzen Czech Republic. E-mail kaisert@. Supported by project 1M0545 and Research Plan MSM 4977751301 of the Czech Ministry of Education. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R38 1 i. If a 1 then any finite coloring of positive integers yields a
