A note on the extensible no-three-in-line problem

Anubhab Ghosal

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arXiv · 2605.07000

A note on the extensible no-three-in-line problem

Anubhab Ghosal

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Abstract

We show the existence of a set $S\subset\mathbb{Z}^2$ avoiding collinear triples satisfying $|S\cap [n]^2|=Ω(n/\sqrt{\log n})$ for sufficiently large $n$. This improves on the best-known lower bound on Erde's extensible no-three-in-line problem due to Nagy, Nagy and Woodroofe by $\sqrt{\log n}$, leaving the same gap to the trivial upper bound. Our construction is random.

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Anubhab Ghosal. 2026-05-07. A note on the extensible no-three-in-line problem. https://arxiv.org/abs/2605.07000

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