Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Two Color Off-diagonal Rado-type Numbers. | Two Color Off-diagonal Rado-type Numbers Kellen Myers kmyers@ Aaron Robertson Department of Mathematics Colgate University Hamilton NY USA aaron@ Submitted Jun 16 2006 Accepted Jul 27 2007 Published Aug 4 2007 Mathematics Subject Classification 05D10 Abstract We show that for any two linear homogeneous equations E0 El each with at least three variables and coefficients not all the same sign any 2-coloring of Z admits monochromatic solutions of color 0 to Eo or monochromatic solutions of color 1 to E1. We define the 2-color off-diagonal Rado number RR E0 E1 to be the smallest N such that 1 N must admit such solutions. We determine a lower bound for RR E0 E1 in certain cases when each Ei is of the form a1x1 . anxn z as well as find the exact value of RR E0 E1 when each is of the form X1 a 2x2 . anxn z. We then present a Maple package that determines upper bounds for off-diagonal Rado numbers of a few particular types and use it to quickly prove two previous results for diagonal Rado numbers. 0 Introduction For r 2 an r-coloring of the positive integers Z is an assignment X Z 0 1 . r 1 . Given a diophantine equation E in the variables x1 . xn we say a solution xi n 1 is monochromatic if x xi x Xj for every i j pair. A well-known theorem of Rado states that for any r 2 a linear homogeneous equation c1 x1 . cnxn 0 with each ci 2 Z admits a monochromatic solution in Z under any r-coloring of Z if and only if some nonempty subset of ci n 1 sums to zero. The smallest N such that any r-coloring of 1 2 . N 1 N satisfies this condition is called the r-color Rado number for the equation E. However Rado also proved the following much lesser known result. Theorem Rado 6 Let E be a linear homogeneous equation with integer coefficients. Assume that E has at least 3 variables with both positive and negative coefficients. Then any 2-coloring of Z admits a monochromatic solution to E. This work was done as part of a summer REU funded by Colgate .
