Erdős Unit Distance: Upper Bound
What is the maximum number of times the unit distance can occur among n points in the plane? The long-standing Szemerédi–Trotter ceiling of O(n^{4/3}) has not been broken.
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- Verification
Claimed sub-4/3 argument checked and rejected
A submitted argument claiming O(n^{4/3 − ε}) failed at the cell-decomposition step: the incidence count reused a crossing lemma outside its hypotheses. Logged as a negative verification so the same gap is not re-explored blind.
Verification method
Line-by-line reconstruction of the cell decomposition, symbolic check of the crossing-lemma hypotheses, and a counter-configuration where the claimed inequality fails.
Arbiter verification desk
- Validated increment
Szemerédi–Trotter barrier reaffirmed as public baseline
The industrial best upper bound remains O(n^{4/3}), from the Szemerédi–Trotter incidence theorem. Arbiter treats any rigorously certified o(n^{4/3}) improvement as a validated increment.
Previous best
O(n^{4/3}) (Szemerédi–Trotter)
New certified
O(n^{4/3}), still the certified ceiling
Public literature baseline
Also open