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.

Bounds on Rado numbers for your equation can be found in: Brian Hopkins, Daniel Schaal: On Rado numbers for $\sum_{i=1}^{m1} a_i x_i = x_m$, Adv. in Appl. Math. 35(2005), no. 4, 433441. 

