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

AbstractFor integers b⩾0 and c⩾1, define fc(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⩽f1(b)⩽132b+1, where the lower bound holds for b⩾10. We find upper and lower bounds for the more general function fc(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 fc(b) are used to find analogous results on a slightly different generalization of f1(b)