An Improved Lower Bound for $S(7)$ and Some Interesting Templates

TL;DR

This paper presents a new lower bound for the Schur number $S(7)$ using a specific partition, and introduces a construction that improves bounds for larger Schur numbers and multicolour Ramsey numbers.

Contribution

The paper provides a new explicit partition that establishes $S(7) ext{ge} 1696$ and introduces a recursive construction to bound larger Schur numbers and Ramsey numbers.

Findings

Established $S(7) ext{ge} 1696$ using a Schur partition.

Derived a recursive inequality $S(k+5) ext{ge} 376.S(k)+160$.

Showed that $ ext{lim}_{r o \infty} R_r(3)^{1/r} > 3.273$.

Abstract

In this simple paper, we exhibit a Schur partition giving rise to a triangle-free linear colouring of $K_{1697}$ in 7 colours. Thus we show that the Schur number $S(7) \ge 1696$ and the multicolour Ramsey number $R_{7}(3) \ge 1698$. We also demonstrate a specific partition of the closed integer interval $[1, 376]$ which, through repeated use of a specific construction, allows us to conclude that $S(k+5) \ge 376.S(k)+160$ for any positive integer $k$. The existence of this partition, with its specific properties, implies that $\lim_{\substack{r \rightarrow \infty}} R{_r}(3)^{1/r} > 3.273$.