Consider the integer linear equation $\sum_{i=1}^{n} c_ix_i=0$ where $c_i(\ne 0) \in Z$. Supposing it is given that there is a natural number N such that, if {1,2...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. 

