The structure of rainbow-free colorings for linear equations on three variables in Zp

Abstract

Let p be a prime number and Z p be the cyclic group of order p. A coloring of Z p is called rainbow-free with respect to a certain equation, if it contains no rainbow solution of the same, that is, a solution whose elements have pairwise distinct colors. In this paper we describe the structure of rainbow-free 3-colorings of Z p 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 Z p contains a rainbow solution of it.

An inverse theorem in $\mathbb{F}_p$ and rainbow free colorings

“…Proof. We have that (i) is a consequence of [6,Lemma 16] up to some cases which are solved easily. Then (ii) is a straightforward consequence of Lemma 2.5.…”

“…The main idea that we will use in the proof is that if then C 1 and C 2 have to be as in (1.2). Due to the main result of [6] and the previous paragraph, we may assume that n > 3. We shall show (7.1) studying the possibilities of m: Suppose that m ≥ 4.…”

“…Jungić et al [8] showed that the inverse theorems are powerful tools to study the rainbow colorings. In the case where n = 3, explicit characterizations of the equations that have rainbow free colorings are provided for example in [8], [9] and [6]. For arbitrary n Conlon [2] showed that under the assumptions min 1≤i≤n |C i | ≥ n and p ≥ 3n 2 − 4n − 3, C is rainbow free with respect to n i=1 a i v i = b only if a 1 = .…”