Rainbow-free colorings for in

Let p be a prime number and Z p be the cyclic group of order p . A 3-coloring of Z p is rainbow-free for some equation if it contains no rainbow solution of the equation. In [3] Jungić et al. (2003) proved that every 3-coloring of Z p , with the cardinality of the smallest color class greater than four, has a rainbow solution of “almost” all linear equations in three variables in Z p . In this work we handle the “small” cases and give a structural description of rainbow-free colorings for the particular case of x + y = c z , which includes the Schur equation.