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Consider the integer linear equation $\sum_{i=1}^{n} c_ix_i=0$, where $c_i(\ne 0) \in \mathbb{Z}$. Supposing it is given that there is a natural number $N$ such that, if $\{1,2, \dots, N\}$ is partitioned in two sets, one of these always contains a solution of the equation. The minimal such $N$ is called the Rado number of the equation. I am looking for general bounds on such $N$, in the cases where it exists. Where can I possibly find such results. Thanks.

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Are you looking for information in terms of the $c_i$? – dvitek Sep 18 '10 at 5:55

Bounds on Rado numbers for your equation can be found in:

Brian Hopkins, Daniel Schaal: On Rado numbers for $\sum_{i=1}^{m-1} a_i x_i = x_m$, Adv. in Appl. Math. 35(2005), no. 4, 433-441.

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Tracing forward through the links at the review of the Hopkins-Schaal paper leads to more recent work, e.g., MR2382523 (2009a:05204), Guo, Song; Sun, Zhi-Wei, Determination of the two-color Rado number for $a_1x_1+\cdots+a_mx_m=x_0$, J. Combin. Theory Ser. A 115 (2008), no. 2, 345–353 and MR2856312, Schaal, Daniel; Zinter, Melanie, Continuous Rado numbers for the equation $a_1x_1+a_2x_2+\cdots+a_{m−1}x_{m−1}+c=x_m$, Proceedings of the Forty-Second Southeastern International Conference on Combinatorics, Graph Theory and Computing, Congr. Numer. 207 (2011), 97–104. – Gerry Myerson Dec 3 '13 at 22:53

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