On some generalizations of the van der Waerden number \(w(3)\) (Q1817569)

The author studies another function related to the two-color three-term van der Waerden number \(w(3)\). He defines \(f(b,c)\) to be the smallest integer \(n\) such that any two-coloring of \(\{ 1,\ldots,n\}\) contains a monochromatic subset of form \(\{ x, x+d, x+2d+b\}\) with some \(d\geq c\). Without that additional parameter \(c\) (just requiring \(d>0\), i.e. \(c=1\)) this function was considered for multiple colors already by \textit{A. Bialostocki, H. Lefmann} and \textit{T. Meerdink} [Discrete Math. 150, No. 1-3, 49-60 (1996; Zbl 0859.05012)], in fact for \(b=0\) it is the classical van der Waerden number. The author gives upper and lower bounds for this function, and for another variation, which improves the earlier bound for the two-color case in the aforementioned paper.