Long Lattice Paths with No Three Collinear Vertices

Abstract

For $d\ge 1$, let $L(d)\in\mathbb N\cup\{\infty\}$ be the supremum of the lengths of paths in $\mathbb Z^d$ whose steps are standard basis vectors and whose vertex sets contain no collinear triple. We prove that \[ \log_2\log_2 L(d)\ge \frac{2}{5}d-O(1) \] for all sufficiently large $d$.

Disclosure

“|I|c(J) · w − |J|c(I) · w = −R(I, J) · w, which is nonzero by (33). Thus the two displacements are not parallel, so no three vertices are collinear. Taking S = {s1 , . . . , sd }, for which |S| ≤ d, proves (29) and (30). Acknowledgments GPT-5.6 Pro was used to assist with optimizing the parameters in the proof. The central amplification construction and the idea of introducing randomness were developed by the author. References [1] S. Avgustinovich and S. Puzynina, Weak abeli”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T05:00:41.949609Z

Structural counts