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North-East Lattice Paths with Few Collinear Vertices

Samuel Korsky

In Korsky’s upper-bound argument for north-east lattice paths with no k collinear visited vertices, the mediant of Farey neighbors raises the uniform density-increment product from 1/[8(k−1)²] to 1/[4(k−1)²]. It gives log A(k)≤(2/e+o(1))(k−1)².

Sources for this record

What was already known

Gerver and Gerver–Ramsey gave weaker upper bounds. Korsky’s density-increment framework already improved the order of the upper bound without the mediant refinement.

AI involvement AI-centered

GPT-5.5 Pro suggested the Farey mediant. Korsky designed the density-increment and random lower-bound methods and proved the refined lemma.

The model supplied a retained mathematical choice central to the specific improved leading constant; the broader asymptotic methods are human-origin.

Importance Modest

Improves the leading constant within the new upper-bound framework.

Limitations

The lower-bound construction and main framework are Korsky’s work. No completed check of the mediant lemma or final asymptotic constant is documented.

Technical verification Reported

A proof is printed, but the author documents computation assistance and responsibility rather than a completed check of this lemma or final bound. Reported is the missing-check default.

Human understanding Full

The author proves Lemma 4.1 by comparing the two Farey endpoints with their mediant and derives the density-increment constant explicitly.

Correction and publication history