Two-color Rado number of x plus y plus c = kz for odd c and k with k >= c+6

Abstract

For positive integers c and k, 2-color Rado number R = R(c, k) is the least integer, provided it exists, such that every 2-coloring of the positive integers up to R admits a monochromatic solution to x + y + c = kz. It is known that R exists if and only if k is odd or c is even. In this paper, for odd c, k we show that R(c, k) = 1/2(k(k + 1) - c + 1) when k >= c + 6. Moreover, we show that 1/2(k(k +1) - c +1) is also an upper bound of R(c, k) when k >= c + 4. (C) 2021 Elsevier B.V. All rights reserved.