2006/06/19 by Myers, Kellen, Robertson, Aaron
#05D10 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.math/0606451
We show that for any two linear homogenous 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.