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The best upper bound, O(n^(4/3)), is due to Spencer, Szemerédi and Trotter (1984).","wording_kind":"paraphrase","locator":"§1.1, History of the problem; §2 opening","supersedes":null}],"reports":[{"id":"erdos-unit-distances:report:false:v1","entry_id":"erdos-unit-distances:entry:remarks:v1","state":"refuted","source_label":"Near-linear Euclidean conjecture refuted","scope":"The n^(1+o(1)) upper bound, not the exact extremal growth.","source_date":"2026-05-20","observed_on":"2026-10-13","evidence_url":"https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-remarks.pdf","supersedes":null},{"id":"erdos-unit-distances:report:gain:v1","entry_id":"erdos-unit-distances:entry:proof:v1","state":"resolved","source_label":"Fixed positive exponent gain proved","scope":"For infinitely many n; unordered unit-distance pairs.","source_date":"2026-05-20","observed_on":"2026-10-13","evidence_url":"https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf","supersedes":null},{"id":"erdos-unit-distances:report:formal:v2","entry_id":"erdos-unit-distances:entry:lean:v2","state":"resolved","source_label":"Formal exponent-gain endpoint reported complete","scope":"Exact Euclidean norm and unordered counting; external proof execution.","source_date":"2026-06-27","observed_on":"2026-10-13","evidence_url":"https://github.com/plby/Erdos90/tree/2062b0e6c9770c81e397bfe41148e90b92ca0567","supersedes":"erdos-unit-distances:report:formal:v1"},{"id":"erdos-unit-distances:report:upper:v1","entry_id":"erdos-unit-distances:entry:remarks:v1","state":"resolved","source_label":"Classical 4/3 upper bound","scope":"Historical universal upper bound; not a claim of optimality.","source_date":null,"observed_on":"2026-10-13","evidence_url":"https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-remarks.pdf","supersedes":null},{"id":"erdos-unit-distances:report:grid:v1","entry_id":"erdos-unit-distances:entry:remarks-history:v1","state":"resolved","source_label":"Erdős’s grid construction","scope":"Only the exact statement and passage cited here; not an independent OpenMath assessment.","source_date":null,"observed_on":"2026-10-21","evidence_url":"https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-remarks.pdf","supersedes":null}]},"status":{"state":"refuted","label":"Resolved · counterexample","extent":"full_statement","summary":"A fixed positive power gain refutes the near-linear Euclidean conjecture in the cited proof and human companion exposition. Exact extremal growth remains a related research question.","basis":"editorial","attributed_to":"OpenMath pilot collection · OpenMath source curation","as_of":"2026-10-13","progress_label":null,"progress_claim_ids":[],"remaining":null,"views":["resolved"],"knowledge_categories":["Related advances"]}},"selected_context":null}}