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arXiv · 2607.02832

North-East Lattice Paths with Few Collinear Vertices

Samuel Korsky
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Abstract

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) gave lower and upper bounds on $A(k)$ of the form \[ \exp\left(Ω(\log(k)^2)\right)\le A(k)\le \exp\left(O(k^4)\right). \] Improving upon these results, we show that \[ \exp\left(Ω(k^{1/3})\right)\le A(k)\le \exp\left(\left(\frac{2}{e}+o(1)\right)(k-1)^2\right). \]

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Samuel Korsky. 2026-07-02. North-East Lattice Paths with Few Collinear Vertices. https://arxiv.org/abs/2607.02832

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