Introduces the extensible no-(k(n)+1)-in-line problem on infinite grids, constructs optimal sets for linear k(n) and positive-density sets for power k(n), proves any high-density configuration requires k(n) growing polynomially, and reduces the constant-k case to regular functions.
A note on the extensible no-three-in-line problem
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abstract
We show the existence of a set $S\subset\mathbb{Z}^2$ avoiding collinear triples satisfying $|S\cap [n]^2|=\Omega(n/\sqrt{\log n})$ 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.
fields
math.CO 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
Any point set P in R^2 has a subset P' with |P'| ≫ |P|^{1/3} in which all distances are distinct.
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The extensible no-$(k(n)+1)$-in-line problem
Introduces the extensible no-(k(n)+1)-in-line problem on infinite grids, constructs optimal sets for linear k(n) and positive-density sets for power k(n), proves any high-density configuration requires k(n) growing polynomially, and reduces the constant-k case to regular functions.
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Geometric Sidon Problems
Any point set P in R^2 has a subset P' with |P'| ≫ |P|^{1/3} in which all distances are distinct.