The 2-color Rado number of x1+x2+...+xm-1=axm
Abstract
In 1982, Beutelspacher and Brestovansky proved that for every integer m≥ 3, the 2-color Rado number of the equation x1+x2+...+xm-1=xm is m2-m-1. In 2008, Schaal and Vestal proved that, for every m≥ 6, the 2-color Rado number of x1+x2+...+xm-1=2xm is m-12m-12. Here we prove that, for every integer a≥ 3 and every m≥ 2a2-a+2, the 2-color Rado number of x1+x2+...+xm-1=axm is m-1am-1a. For the case a=3, we show that our formula gives the Rado number for all m≥ 7, and we determine the Rado number for all m≥ 3.
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