PreprintCC BY 4.0arXiv
A new lower bound for the sixth Schur number: S(6)≥ 537
Abstract
We give an explicit partition of {1,…,537} into six sum-free sets, establishing S(6)≥ 537. This improves the lower bound 536 of Fredricksen and Sweet (2000). Known recursive constructions then give the improved lower bounds S(8)≥ 5 372, S(11)≥ 204 208, S(13)≥ 2 042 082, and R6(3)≥ 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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BibTeX
@misc{chen2026lower,
title = {{A new lower bound for the sixth Schur number: $S(6)\ge 537$}},
author = {Zhiyang Chen and Ligeng Zhu and Fan Long},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31362},
eprint = {2610.31362},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31362},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A new lower bound for the sixth Schur number: S(6)≥ 537
AU - Zhiyang Chen
AU - Ligeng Zhu
AU - Fan Long
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31362
UR - https://arxiv.org/abs/2610.31362
AB - We give an explicit partition of {1,…,537} into six sum-free sets, establishing S(6)≥ 537. This improves the lower bound 536 of Fredricksen and Sweet (2000). Known recursive constructions then give the improved lower bounds S(8)≥ 5 372, S(11)≥ 204 208, S(13)≥ 2 042 082, and R_6(3)≥ 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.
ER -