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Points and plane unit and distance
You likely mean the Erdős unit distance problem . There is no known full proof of Erdős’s conjecture; it remains open in the sense that the best known upper bound is still far above the conjectured bound [2][5].
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You likely mean the Erdős unit distance problem. There is no known full proof of Erdős’s conjecture; it remains open in the sense that the best known upper bound is still far above the conjectured bound .
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The problem asks: among (n) points in the Euclidean plane, what is the maximum possible number of pairs exactly distance 1 apart?
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Erdős conjectured that the true maximum is almost linear, roughly (n^{1+o(1)}), based on lattice examples giving many unit distances
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The best general upper bound known is still (O(n^{4/3})), proved by Spencer, Szemerédi, and Trotter in 1984
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So the current situation is
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What is the short answer to "Points and plane unit and distance"?
You likely mean the Erdős unit distance problem .
What are the key points to validate first?
You likely mean the Erdős unit distance problem . There is no known full proof of Erdős’s conjecture; it remains open in the sense that the best known upper bound is still far above the conjectured bound [2][5].
What should I do next in practice?
The problem asks: among \(n\) points in the Euclidean plane, what is the maximum possible number of pairs exactly distance 1 apart?
Sources
- arxiv.orgErdős's unit distance problem and rigidity - arXiv
- erdosproblems.comn - Erdős Problems
- youtube.comErd\H{o}s Unit Distance Problem and Graph Rigidity - Orit Raz
- arxiv.orgarXiv:2302.09058v2 [math.CO] 7 Nov 2024
- math.uchicago.eduERD¨OS DISTANCE PROBLEMS
- arxiv.orgThe Erdős unit distance problem for small point sets - arXiv
- en.wikipedia.orgErdős distinct distances problem - Wikipedia
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