Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Generalized Schur Numbers for x1 + x2 + c = 3x3. | Generalized Schur Numbers for x1 x2 c 3x3 Andre E. Kezdy Department of Mathematics University of Louisville Louisville KY 40292 USA kezdy@ Hunter S. Snevily Department of Mathematics University of Idaho Moscow ID 83844 USA snevily@ Susan C. White Department of Mathematics University of Louisville Louisville KY 40292 USA Submitted Jul 10 2008 Accepted Jul 30 2009 Published Aug 14 2009 Mathematics Subject Classification 05D10 Abstract Let r c be the least positive integer n such that every two coloring of the integers 1 . n contains a monochromatic solution to x1 x2 c 3x3. Verifying a conjecture of Martinelli and Schaal we prove that r c 21 2y 1 c 3 for all c 13 and r c 3T w1 - c 2 for all c -4. Section 1. Introduction Let N denote the set of positive integers and a b n G N a n b . A map X a b 1 t is a t-coloring of a b . Let L be a system of equations in the variables x1 . xm. A positive integral solution n1 . nm to L is monochromatic if x nf x nj for all 1 i j m. The t-color generalized Schur number of L denoted St L is the least positive integer n if it exists such that any t-coloring of 1 n results in a monochromatic solution to L. If no such n exists then St L is X. THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R105 1 2 2f Ị c -3 that 3 Ị- c-2 A classical result of Schur 5 states that St L TO for L x1 x2 x3 and all t 2. An exercise is to show that S4 L TO for L x y 3z . Very few generalized Schur numbers are known but several recent papers have revived interest in determining some of them for example 1 2 3 4 . In this paper we answer a conjecture posed by Martinelli and Schaal 3 concerning the 2-color generalized Schur number of the equation X1 x2 c 3x3. This number is denoted r c . Verifying the conjecture we prove in section that r c 1 3 for all c 13 and we prove in section that r c pr-b for all c -4. Martinelli and Schaal were motivated to consider a more general equation X1 x2 c kx3 where c is an arbitrary
