Rainbow-free colorings for in

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