Sets avoiding a rainbow solution to the generalized Schur equation

2025/06/20 by Győri, Ervin, He, Zhen, Lv, Zequn +4
Mathematics · #05D99 #Advanced Combinatorial Mathematics #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2506.17117

openalex publication_date 2025/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 A₁,A₂,…,A_k ⊆ [n] avoiding a rainbow solution to the Schur equation x₁+x₂+…+x_m=x_m+1. Moreover, we determine all the extremal families.

Citations

• Rainbow Numbers for the Generalized Schur Equation x₁ + x₂ + ⋯ + x_m-1 = x_m
• Extremal results for graphs avoiding a rainbow subgraph
• On solution-free sets of integers II
• Extremal subsets of 1,...,n avoiding solutions to linear equations in three variables
• On Certain Sets of Integers

Related

• Rainbow Numbers for the Generalized Schur Equation x₁ + x₂ + ⋯ + x_m-1 = x_m
• On product Schur triples in the integers
• Gallai-Schur Triples and Related Problems
• Rainbow numbers of [m] × [n] for x₁ + x₂ = x₃
• Schur-like numbers and a lemma of Shearer