A construction for weak Schur partitions

TL;DR

This paper introduces an iterative construction method that provides a new lower bound for weak Schur numbers, improving upon Braun's earlier formula and contributing to the understanding of weak Schur partitions.

Contribution

The paper presents a novel iterative construction that yields a formulaic lower bound for weak Schur numbers, extending previous results and offering a new approach to weak Schur partitions.

Findings

Reproduces the known bound WS(6) ≥ 554

Exceeds Braun's bound for all larger s

Provides a new construction method for weak Schur partitions

Abstract

In 1952, J.H.Braun claimed to have established a formula giving a lower bound for certain partitions of sets of integers into weakly sum-free classes. However, no proof or supporting construction was published at that time. In today's terminology, that claim was equivalent to giving a formulaic lower bound for the weak Schur number $WS(s)$. $WS(s)$ is the maximum number such that there exists a weak Schur partition of the integers from 1 to $WS(s)$, into $s$ subsets. In a weak Schur partition of a set of integers, there can be no three distinct members $a$, $b$ and $c$ in any subset, such that $a+b=c$. An iterative construction described in this paper results in a similar formulaic lower bound. Although different from that given by Braun, it reproduces the result $WS(6) \ge 554$ implied by his formula, and exceeds it for all larger values of $s$. Various starting points can be used as a basis for the iterations. This result itself is no longer remarkable: it has been proven elsewhere that $WS(6) \ge 642$. Even so, it is hoped that the formula and its underlying construction may nevertheless be of interest to those interested in weak Schur partitions and/or the closely-related linear Ramsey graphs.