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For all positive integers  s  and  t , Brown et al. defined    to be the smallest positive integer  N  such that every 2-coloring of    has a monochromatic homothetic copy of {1,1+ s ,1+ s + t }. They proved that    for all  s ,  t  and that equality holds in the case where both    and    with    and in many other cases. They also proved that for all positive integers  m ,    or   . In this paper, we show that    and that for all the other ( s , t ),   .

Van der Waerden's Theorem on Homothetic Copies of {1,1+ s ,1+ s + t }