{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:north-east-lattice-paths-with-few-collinear-vertices", "title": "North-East Lattice Paths with Few Collinear Vertices", "canonical_statement": "Let $A(k)$ be the largest possible number of moves in a north-east lattice path whose visited vertices contain no $k$ collinear points. Gerver (1979) and Gerver and Ramsey (1979) bounded $A(k)$ by\n$$\\exp\\left(\\Omega\\left(\\log(k)^2\\right)\\right) \\le A(k) \\le \\exp\\left(O\\left(k^4\\right)\\right),$$\nand determining the true growth rate has been open since. Both bounds are improved to\n$$\\exp\\left(\\Omega\\left(k^{1/3}\\right)\\right) \\le A(k) \\le \\exp\\left(O\\left(k^2\\right)\\right),$$\nwith the upper bound proved in the sharper form $\\exp\\left(\\left(\\tfrac{2}{e}+o(1)\\right)(k-1)^2\\right)$.", "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.", "current_status": "partial", "solution_events": [ { "id": "vibemathed:north-east-lattice-paths-with-few-collinear-vertices:event", "problem_id": "vibemathed:north-east-lattice-paths-with-few-collinear-vertices", "type": "proved", "status": "partial", "occurred_at": "2026-07-02T00:00:00.000Z", "title": "North-East Lattice Paths with Few Collinear Vertices (arXiv)", "summary": "Both bounds move, and the gap stays enormous: the lower bound rises from $\\exp(\\Omega(\\log^2 k))$ to $\\exp(\\Omega(k^{1/3}))$ and the upper falls from $\\exp(O(k^4))$ to $\\exp(O(k^2))$, so $A(k)$ is still undetermined between an exponent of $k^{1/3}$ and one of $k^2$. The paper's own closing discussion argues its lower-bound construction is near the limit of the method and that beating it needs additional randomness, a sharper line-counting step, or a different model entirely.", "ai_contribution": "ai_assisted", "attempt_ids": [ "vibemathed:north-east-lattice-paths-with-few-collinear-vertices:attempt" ], "method_family_ids": [ "vibemathed:north-east-lattice-paths-with-few-collinear-vertices:method" ], "source_assertion_ids": [ "vibemathed:north-east-lattice-paths-with-few-collinear-vertices:assertion" ] } ], "attempts": [ { "id": "vibemathed:north-east-lattice-paths-with-few-collinear-vertices:attempt", "problem_id": "vibemathed:north-east-lattice-paths-with-few-collinear-vertices", "model": "GPT-5.5 Pro", "provider": "OpenAI", "attempted_at": "2026-07-02T00:00:00.000Z", "human_collaborators": [ "Samuel Korsky" ], "exposure": "unknown", "prompt_public": null, "outcome": "The acknowledgement in full: the author was assisted by GPT-5.5 Pro in preparing the paper, but \"the main construction ideas, including the dyadic-interval random variables in the lower bound and the density-increment framework in the upper bound, were due to the author\". AI tools checked computations, assisted with drafting, and improved the upper-bound constant by suggesting the use of the mediant of the relevant Farey fractions. That last contribution is traceable in the text: it lifts the density increment from $(1/8-o(1))(k-1)^{-2}$ to $(1/4-o(1))(k-1)^{-2}$, which is what produces the $2/e$ constant. So the model sharpened the constant inside the new upper bound rather than the exponent, which is the lower tier by this site's definition.", "failed_routes": [], "artifacts": [ { "label": "Gerver, Long walks in the plane with few collinear points (Pacific J. Math. 83, 1979)", "url": "https://doi.org/10.2140/pjm.1979.83.349", "kind": "problem-record", "license": null }, { "label": "Gerver and Ramsey, On certain sequences of lattice points (Pacific J. Math. 83, 1979)", "url": "https://doi.org/10.2140/pjm.1979.83.357", "kind": "problem-record", "license": null }, { "label": "Lidbetter, Improved bound for the Gerver-Ramsey collinearity problem (Discrete Math., 2024)", "url": "https://doi.org/10.1016/j.disc.2023.113718", "kind": "paper", "license": null }, { "label": "Barnoff and Bright, North-East lattice paths avoiding k collinear points via satisfiability (2026)", "url": "https://doi.org/10.1016/j.aam.2026.103112", "kind": "paper", "license": null } ], "sources": [ { "label": "North-East Lattice Paths with Few Collinear Vertices (arXiv)", "url": "https://arxiv.org/abs/2607.02832", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: North-East Lattice Paths with Few Collinear Vertices", "url": "https://vibemathed.com/problem/north-east-lattice-paths-with-few-collinear-vertices", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:north-east-lattice-paths-with-few-collinear-vertices:verification", "problem_id": "vibemathed:north-east-lattice-paths-with-few-collinear-vertices", "solution_event_id": "vibemathed:north-east-lattice-paths-with-few-collinear-vertices:event", "level": "unreviewed", "mathematical_correctness": "unknown", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2607.02832, Korsky, 2 July 2026). The abstract matches this entry, and the acknowledgement is verbatim as the AI-role note now quotes it - including the sentence attributing the main construction ideas to the author, which the submission's quote had omitted. The model's named contribution was traced through the text to the density-increment step it actually improves. The prior bounds attribute correctly: Gerver, Pacific J. Math. 83 (1979) 349-355, and Gerver-Ramsey, same volume, 357-363. The proofs themselves - a dyadic slope-field random construction and a Farey-mediant density increment - were not checked here and need a discrete geometer. Unrefereed preprint, no independent review.", "sources": [ { "label": "VibeMathed: North-East Lattice Paths with Few Collinear Vertices", "url": "https://vibemathed.com/problem/north-east-lattice-paths-with-few-collinear-vertices", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "North-East Lattice Paths with Few Collinear Vertices (arXiv)", "url": "https://arxiv.org/abs/2607.02832", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "North-East Lattice Paths with Few Collinear Vertices (arXiv)", "url": "https://arxiv.org/abs/2607.02832", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: North-East Lattice Paths with Few Collinear Vertices", "url": "https://vibemathed.com/problem/north-east-lattice-paths-with-few-collinear-vertices", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "expand", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:north-east-lattice-paths-with-few-collinear-vertices:signal:recent_partial_result", "problem_id": "vibemathed:north-east-lattice-paths-with-few-collinear-vertices", "kind": "recent_partial_result", "reason": "A source-reported partial result may support bounded expansion.", "generated_at": "2026-09-13T16:28:40.178Z" } ], "uncertainties": [ "VibeMath has not independently verified the mathematical claim.", "AI-attempt independence and training-data exposure are unknown unless explicitly documented.", "VibeMath has not independently audited the mathematical statement, proof, or novelty claim." ], "generated_at": "2026-09-13T16:28:40.178Z" }