Rainbow Solutions to the Sidon Equation in Cyclic Groups

TL;DR

This paper investigates the minimum number of colors needed in cyclic groups to guarantee rainbow solutions to the Sidon equation, providing new results for prime and general cyclic groups.

Contribution

It proves that every 4-coloring of prime cyclic groups contains a rainbow Sidon solution and determines the rainbow number for all cyclic groups.

Findings

Every 4-coloring of inite cyclic groups with prime order contains a rainbow solution.

The rainbow number for the Sidon equation in cyclic groups is explicitly determined.

Results extend understanding of rainbow solutions in additive combinatorics.

Abstract

Given a coloring of group elements, a rainbow solution to an equation is a solution whose every element is assigned a different color. The rainbow number of $\mathbb{Z}_n$ for an equation $eq$, denoted $rb(\mathbb{Z}_n,eq)$, is the smallest number of colors $r$ such that every exact $r$-coloring of $\mathbb{Z}_n$ admits a rainbow solution to the equation $eq$. We prove that for every exact $4$-coloring of $\mathbb{Z}_p$, where $p\geq 3$ is prime, there exists a rainbow solution to the Sidon equation $x_1+x_2=x_3+x_4$. Furthermore, we determine the rainbow number of $\mathbb{Z}_n$ for the Sidon equation.