A local-breadth search algorithm generates unit-distance graphs in the rational plane whose scaling exponent exceeds recent theoretical lower bounds.
An explicit lower bound for the unit distance problem
4 Pith papers cite this work. Polarity classification is still indexing.
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Pith papers citing it
abstract
We show that there are sets of $n$ points in the plane with $n$ arbitrarily large that contain more than $n^{1.014}$ pairs of points separated by a distance exactly $1$. This improves on very recent work of a team at OpenAI, who proved the same result with an inexplicit exponent greater than $1$, drastically improving on the best previous lower bound and disproving a conjecture of Erd\H{o}s. The method is number-theoretic, relying on constructing algebraic number fields of large degree and small discriminant with many primes of small norm via a Golod-Shafarevich criterion argument.
years
2026 4verdicts
UNVERDICTED 4representative citing papers
Any point set P in R^2 has a subset P' with |P'| ≫ |P|^{1/3} in which all distances are distinct.
Optimizes integer multiplicities and a real parameter in Sawin's unit-distance certificate via an evolution strategy to obtain improved explicit lower bounds u(n) > n^{1.0152} and u(n) > n^{1.031}.
An incomplete attempt to show that the O(n^{4/3}) upper bound for unit distances among n points in the plane is not sharp, plus remarks on Szemerédi-Trotter incidences.
citing papers explorer
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Algorithmic exploration of the unit distance problem in the rational plane
A local-breadth search algorithm generates unit-distance graphs in the rational plane whose scaling exponent exceeds recent theoretical lower bounds.
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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.
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Optimizing Explicit Unit-Distance Lower-Bound Certificates
Optimizes integer multiplicities and a real parameter in Sawin's unit-distance certificate via an evolution strategy to obtain improved explicit lower bounds u(n) > n^{1.0152} and u(n) > n^{1.031}.
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An incomplete attack on the upper bound of the unit distance problem
An incomplete attempt to show that the O(n^{4/3}) upper bound for unit distances among n points in the plane is not sharp, plus remarks on Szemerédi-Trotter incidences.