Summary: Van der Waerden’s classical theorem on arithmetic progressions states that for any positive integers \(k\) and \(r\), there exists a least positive integer, \(w(k,r)\), such that any \(r\)-coloring of \(\{1,2,\dots,w(k,r)\}\) must contain a monochromatic \(k\)-term arithmetic progression \(\{x,x+d,x+ 2d,\dots,x+ (k-1)d\}\). We investigate the following generalization of \(w(3,r)\). For fixed positive integers \(a\) and \(b\) with \(a\leq b\), define \(N(a,\,b;r)\) to be the least positive integer, if it exists, such that any \(r\)-coloring of \(\{1,2,\dots, N(a,b;r)\}\) must contain a monochromatic set of the form \(\{x,ax+ d,\,bx+ 2d\}\). We show that \(N(a,b;2)\) exists if and only if \(b\neq 2a\), and provide upper and lower bounds for it. We then show that for a large class of pairs \((a,b)\), \(N(a,\,b;r)\) does not exist for \(r\) sufficiently large. We also give a result on sets of the form \(\{x,ax+ d,ax+ 2d,\dots, ax+ (k-1)d\}\).