{"paper":{"title":"Elementary incidence theorems for complex numbers and quaternions","license":"","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Jozsef Solymosi, Konrad J. Swanepoel","submitted_at":"2007-03-13T10:55:57Z","abstract_excerpt":"We present some elementary ideas to prove the following Sylvester-Gallai type theorems involving incidences between points and lines in the planes over the complex numbers and quaternions.\n  (1) Let A and B be finite sets of at least two complex numbers each. Then there exists a line l in the complex affine plane such that l intersects AxB in exactly two points.\n  (2) Let S be a finite noncollinear set of points in the complex affine plane. Then there exists a line l intersecting S in 2, 3, 4 or 5 points.\n  (3) Let A and B be finite sets of at least two quaternions each. Then there exists a li"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0703372","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0703372/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}