Metraipolis › archive › dc_38759d477ee2

Sparse coloring for isosceles-free grids: an AI-assisted proof case study — arXiv:2610.04194

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.04194
License (as archived): CC0-1.0
Archived: 2026-10-06T16:51:29.956Z
SHA-256 of the archived text: b9ac4b8b98a6640558f62c79114628607ec87d058f6839319f7e533cd15a9e76
Stable URL of this page: https://metraipolis.com/documento/dc_38759d477ee2

How to cite

Sparse coloring for isosceles-free grids: an AI-assisted proof case study — arXiv:2610.04194 — arXiv (en), https://arxiv.org/abs/2610.04194 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_38759d477ee2 (Metraipolis archive dc_38759d477ee2, sha256 b9ac4b8b98a6640558f62c79114628607ec87d058f6839319f7e533cd15a9e76)

BibTeX

@misc{dc38759d477ee2,
  title = {Sparse coloring for isosceles-free grids: an AI-assisted proof case study — arXiv:2610.04194},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.04194},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_38759d477ee2; sha256 b9ac4b8b98a6640558f62c79114628607ec87d058f6839319f7e533cd15a9e76},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.04194. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We study the largest size $C(n)$ of a subset of the $n\times n$ integer grid containing no isosceles triple, including equally spaced collinear triples. We prove $C(n)=\Omega\left(n\sqrt{\log\log n/\log n}\right)$ by applying the sparse hypergraph coloring theorem of Cooper and Mubayi. Elementary geometry and primitive-direction counts give maximum degree $O(n^2\log n)$ and pair-codegree at most $5n$ for the forbidden-triple hypergraph. The resulting bound improves the explicit guarantee in PatternBoost by a factor of order $\sqrt{\log\log n}$. Recovered interaction records document how a Codex literature proposal, human route selection, finite verification, and author-relayed review informed proof synthesis and repair. We connect these actions to successive proof artifacts and extract two case-derived checking practices: tracking objects, parameters and conclusions in theorem applications, and seeding a definitional omission to test a finite verifier. Separately implemented enumerators agree on complete edge sets for every $2\le n\le12$; suppressing the degenerate-case branch loses exactly the equally spaced collinear triples. These records document assistance under human direction, and finite checks support implementation consistency rather than the asymptotic theorem itself.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2610.04194 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-06T15:19:19.967Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).