Sets avoiding a rainbow solution to the generalized Schur equation

TL;DR

This paper extends classical combinatorial number theory results to multicolored sets avoiding rainbow solutions to generalized Schur equations, identifying extremal families and maximizing combined set sizes.

Contribution

It introduces multicolored extensions of Schur equation avoidance, determining extremal families and maximizing sum and product of set sizes.

Findings

Maximized sum and product of set sizes avoiding rainbow solutions.

Identified all extremal families for the problem.

Extended classical results to multicolored and generalized equations.

Abstract

A classical result in combinatorial number theory states that the largest subset of $[n]$ avoiding a solution to the equation $x+y=z$ is of size $\lceil n/2 \rceil$. For all integers $k>m$, we prove multicolored extensions of this result where we maximize the sum and product of the sizes of sets $A_1,A_2,\dots,A_k \subseteq [n]$ avoiding a rainbow solution to the Schur equation $x_1+x_2+\dots+x_m=x_{m+1}$. Moreover, we determine all the extremal families.