On some generalizations of the van der Waerden number w(3)

For integers b≥0 and c≥ 1, define f_c(b) to be the least positive integer n such that for every 2-coloring of [1, n] there is a monochromatic sequence of the form {x, x + d, x + 2d + b} where x and d are positive integers with d ≥c. Bialostocki, Lefmann, and Meerdink showed that for b even, 2b + 10≤f⊥(b)≤13/2b + 1, where the lower bound holds for b≥10. We find upper and lower bounds for the more general function f_c(b) which, for c = 1, improve the aforementioned upper bound to ⌈9b/4⌉ + 9, and give the same lower bound of 2b + 10. Results about f_c(b) are used to find analogous results on a slightly different generalization of f⊥(b).