North-East Lattice Paths with Few Collinear Vertices
partialconfidence 70%
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Precise 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
$$\exp\left(\Omega\left(\log(k)^2\right)\right) \le A(k) \le \exp\left(O\left(k^4\right)\right),$$
and determining the true growth rate has been open since. Both bounds are improved to
$$\exp\left(\Omega\left(k^{1/3}\right)\right) \le A(k) \le \exp\left(O\left(k^2\right)\right),$$
with the upper bound proved in the sharper form $\exp\left(\left(\tfrac{2}{e}+o(1)\right)(k-1)^2\right)$.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
GPT-5.5 Pro
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.
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.
North-East Lattice Paths with Few Collinear Vertices (arXiv)
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.
Known method families
argument (source-reported)
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
What remains uncertain
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VibeMath has not independently audited the mathematical statement, proof, or novelty claim.