THE STRUCTURE OF RAINBOW-FREE COLORINGS FOR LINEAR EQUATIONS ON THREE VARIABLES IN Zp

摘要

Let p be a prime number and Zp be the cyclic group of order p. A coloring of Zp is called rainbow–free with respect to a certain equation, if it contains no rainbow solution of the equation, that is, a solution whose elements have pairwise distinct colors. In this paper we describe the structure of rainbow–free 3–colorings of Zp with respect to all linear equations on three variables. Consequently, we determine those linear equations on three variables for which every 3–coloring (with nonempty color classes) of Zp contains a rainbow solution of it.