Length bounds for lattice paths avoiding k collinear vertices
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Let A(k) bound the length of north-east lattice paths with no k collinear visited vertices. The target is a sharper density-increment argument giving log A(k)≤(2/e+o(1))(k−1)², with uniform product 1/[4(k−1)²].
Why it matters and what counts as progress
Why it matters: It seeks a better leading constant in the quantitative obstruction to long paths avoiding collinearity.
What counts as progress: Sharper density increments and leading constants for the maximum length of a north-east path avoiding k collinear visited vertices.
How the problem developed
AI contribution
North-East Lattice Paths with Few Collinear Vertices
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.
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