On generalized Schur numbers for x1+x2+c = kx3

For every integer c and every positive integer k, let n = r(c, k) be the least integer, provided that it exists, such that for every coloring Delta: {1, 2,..., n} -> {0, 11}, there exist three integers, x(1), x(2), x(3), (not necessarily distinct) such that Delta(x(1)) = Delta(x(2)) = Delta(x(3)) and x(1) + x(2) + c = kx(3) If such an integer does not exist, then let r(c, k) = infinity. The main result of this paper is that [GRAPHICS] for every integer c. In addition, a lower bound is found for r(c, k) for all integers c and positive integers k and linear upper and lower bounds are found for r(c, 3) for all positive integers c.