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

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 ${\lim }_{\begin{array}{c}r\to \infty \end{array}}R{}_r(3{)}^{1/r}>3.273$.

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