Two color off-diagonal Rado-type numbers
The electronic journal of combinatorics, Tome 14 (2007)
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Zbl arXiv EuDML
We show that for any two linear homogeneous equations ${\cal E}_0,{\cal E}_1$, each with at least three variables and coefficients not all the same sign, any 2-coloring of ${\Bbb Z}^+$ admits monochromatic solutions of color 0 to ${\cal E}_0$ or monochromatic solutions of color 1 to ${\cal E}_1$. We define the 2-color off-diagonal Rado number $RR({\cal E}_0,{\cal E}_1)$ to be the smallest $N$ such that $[1,N]$ must admit such solutions. We determine a lower bound for $RR({\cal E}_0,{\cal E}_1)$ in certain cases when each ${\cal E}_i$ is of the form $a_1x_1+\dots+a_nx_n=z$ as well as find the exact value of $RR({\cal E}_0,{\cal E}_1)$ when each is of the form $x_1+a_2x_2+\dots+a_nx_n=z$. 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.
Kellen Myers; Aaron Robertson. Two color off-diagonal Rado-type numbers. The electronic journal of combinatorics, Tome 14 (2007). doi: 10.37236/971
@article{10_37236_971,
author = {Kellen Myers and Aaron Robertson},
title = {Two color off-diagonal {Rado-type} numbers},
journal = {The electronic journal of combinatorics},
year = {2007},
volume = {14},
doi = {10.37236/971},
zbl = {1157.05336},
url = {http://geodesic.mathdoc.fr/articles/10.37236/971/}
}
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