Modular Schur numbers

TL;DR

This paper determines the exact values of generalized modular Schur numbers for all positive integers k and l when m equals 1, 2, and 3, extending the understanding of sum-free partitions in modular arithmetic.

Contribution

The paper explicitly calculates the generalized modular Schur numbers for m=1, 2, and 3, providing new exact results in the theory of sum-free partitions.

Findings

Exact values of modular Schur numbers for m=1, 2, 3

Extension of sum-free partition theory in modular settings

Provides foundational results for further research in modular combinatorics

Abstract

For any positive integers l and m, a set of integers is said to be (weakly) l-sum-free modulo m if it contains no (pairwise distinct) elements $x_1,x_2,...,x_l,y$ satisfying the congruence $x_1+\...+x_l\equiv y\bmod{m}$. It is proved that, for any positive integers k and l, there exists a largest integer $n$ for which the set of the first $n$ positive integers $\{1,2,\...,n\}$ admits a partition into k (weakly) l-sum-free sets modulo m. This number is called the generalized (weak) Schur number modulo $m$, associated with k and l. In this paper, for all positive integers k and l, the exact value of these modular Schur numbers are determined for m=1, 2 and 3.