We give an explicit partition of $\{1,\ldots,537\}$ into six sum-free sets, establishing $S(6)\ge 537$. This improves the lower bound 536 of Fredricksen and Sweet (2000). Known recursive constructions then give the improved lower bounds $S(8)\ge 5\,372$, $S(11)\ge 204\,208$, $S(13)\ge 2\,042\,082$, and $R_6(3)\ge 539$. We also report 131 pairwise inequivalent partitions found by the search. We describe their structure and provide the complete partitions in the supplementary material. These partitions provide starting points for future searches and structural studies of Schur partitions.

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