Fredricksen and Sweet published a sum-free six-color partition of the integers 1 through 536, proving the lower bound S(6) ≥ 536. This repository checks their certificate and records a separate, local observation about extending that specific labeled coloring.
Let c be the six-coloring transcribed from their paper. If a sum-free
six-coloring d of 1 through 537 exists, then d must assign a different
color from c at at least 34 positions among 1 through 536. The labels of
the six colors are held fixed; the distance is not minimized over color
permutations. This does not say whether such a d exists, give an exact
minimum distance, determine S(6), or improve the published global bound.
- Research page in English, German, and Simplified Chinese, with both reproduction ZIPs and the earlier website versions
- English manuscript and PDF reading copy
- German summary and Simplified Chinese summary
- LaTeX source and PDF build script
The German and Chinese files are summaries. The full proof and its 20-triple
table are in the English manuscript. The radius33 file names refer to the
exclusion of all extensions at distance at most 33.
From the repository root, with Python 3 and no nonstandard packages for the mathematical checks:
python repro/verify.py
python repro/exact32.py
python repro/exact33.py
python repro/audit33.pyThe scripts check the 536-entry certificate, all 71,824 Schur triples with
1 ≤ x ≤ y and x+y ≤ 536, the complementary-pair counts for 537, and the
20 explicit blocker triples used in the local proof. Expected results are in REPRODUCIBILITY.md.
MANIFEST.sha256 records the original release snapshot; its README hash belongs to that snapshot rather than this current overview. The independently documented comparison between the JSON lists and the original printed lists is in source-comparison.md.
The 536-color certificate and the bound S(6) ≥ 536 are due to Harold Fredricksen and Melvin M. Sweet, “Symmetric Sum-Free Partitions and Lower Bounds for Schur Numbers,” The Electronic Journal of Combinatorics 7 (2000), R32. Our JSON file is a transcription of their six published lists, expanded by the scripts using the symmetry and exception stated in that paper. The local 34-change argument, accompanying scripts, and note are attributed to Beedbyte. Correspondence: beedbyte@3g-projects.de.
AI agents transcribed the published partition, wrote and ran the finite checkers, and performed internal checks of the local argument. A separate source comparison checked all 269 printed certificate entries against the publisher's PDF. The explicit witnesses and deterministic checks are supplied for independent review. This work has not undergone external peer review. No DOI has been assigned to this repository package.