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Two Color Off-diagonal Rado-type Numbers

By Kellen Myers and Aaron Robertson


We show that for any two linear homogeneous equations E0, E1, each with at least three variables and coefficients not all the same sign, any 2-coloring of Z + admits monochromatic solutions of color 0 to E0 or monochromatic solutions of color 1 to E1. We define the 2-color off-diagonal Rado number RR(E0, E1) to be the smallest N such that [1, N] must admit such solutions. We determine a lower bound for RR(E0, E1) in certain cases when each Ei is of the form a1x1 +... + anxn = z as well as find the exact value of RR(E0, E1) when each is of the form x1 + a2x2 +... + anxn = z. We then present a Maple package that determines upper bounds for off-diagonal Rado numbers of a few particular types, and use it to quickly prove two previous results for diagonal Rado numbers.

Year: 2010
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