Two and Three-color Rado numbers for x1+x2+⋯+xn=y2

초록

An r-coloring of a set S is a function from S to {0,1,& ctdot;,r-1}. Let (En) be the equation x1+x2+& ctdot;+xn=y2. The r-color Rado number Rr(En) of (En) is the least integer R, provided it exists, such that every r-coloring of {1,2,& ctdot;,R} admits a monochromatic solution to (En). In this study, for each n,r >= 2, we provide a lower bound & ell;r of Rr(En). Accordingly, we show that when 2 <= n <= 6, R2(En)=n, and when n >= 7, R2(En)=nn}, which coincides with the lower bound & ell;2 that we have provided. We also show that R3(En)=n when 2 <= n <= 11, and R3(E12)=11.

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