New lower bounds for Schur and weak Schur numbers

TL;DR

This paper introduces a template-based method to establish new lower bounds for Schur and weak Schur numbers, significantly advancing understanding of their growth rates and related Ramsey numbers.

Contribution

It generalizes the concept of templates to weak Schur numbers and provides explicit partitions that improve known lower bounds and growth estimates.

Findings

New lower bounds for S(9) and S(10)

New lower bounds for WS(6), WS(9), and WS(10)

Improved growth rate estimates for Schur and weak Schur numbers

Abstract

This article provides new lower bounds for both Schur and weak Schur numbers by exploiting a "template"-based approach. The concept of "template" is also generalized to weak Schur numbers. Finding new templates leads to explicit partitions improving lower bounds as well as the growth rate for Schur numbers, weak Schur numbers, and multicolor Ramsey numbers $R_n(3)$. The new lower bounds include $S(9) \geq 17\,803$, $S(10) \geq 60\,948$, $\mathit{WS}(6) \geq 646$, $\mathit{WS}(9) \geq 22\,536$ and $\mathit{WS}(10) \geq 71\,256$.