Rainbow linear equations on three variables in

Let p be a prime number and Z p be the cyclic group of order p. A 3–coloring of Z p is called rainbow–free with respect to a certain equation, if it contains no rainbow solution of the same. We describe the structure of rainbow–free colorings with respect to all linear equations on three variables. Consequently, we determine those linear equations on three variables for which every 3–coloring of Z p with nonempty color classes contains a rainbow solution of it.