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North-East Lattice Paths with Few Collinear Vertices

Combinatorics · posed by Joseph L. Gerver, L. Thomas Ramsey, 1979 · partial

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Statement

Let A(k)A(k) be the largest possible number of moves in a north-east lattice path whose visited vertices contain no kk collinear points. Gerver (1979) and Gerver and Ramsey (1979) bounded A(k)A(k) by exp⁡(Ω(log⁡(k)2))≤A(k)≤exp⁡(O(k4)),\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⁡(Ω(k1/3))≤A(k)≤exp⁡(O(k2)),\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⁡((2e+o(1))(k−1)2)\exp\left(\left(\tfrac{2}{e}+o(1)\right)(k-1)^2\right).

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    Samuel Korsky, using GPT-5.5 Pro

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai assisted
    — a person led the work and used a model along the way.

    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(1/8-o(1))(k-1)^{-2} to (1/4−o(1))(k−1)−2(1/4-o(1))(k-1)^{-2}, which is what produces the 2/e2/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.

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