{"paper":{"title":"Tight Bound and Structural Theorem for Joints","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hung-Hsun Hans Yu, Ting-Wei Chao","submitted_at":"2023-07-28T08:03:58Z","abstract_excerpt":"A joint of a set of lines $\\mathcal{L}$ in $\\mathbb{F}^d$ is a point that is contained in $d$ lines with linearly independent directions. The joints problem asks for the maximum number of joints that are formed by $L$ lines. Guth and Katz showed that the number of joints is at most $O(L^{3/2})$ in $\\mathbb{R}^3$ using polynomial method. This upper bound is met by the construction given by taking the joints and the lines to be all the $d$-wise intersections and all the $(d-1)$-wise intersections of $M$ hyperplanes in general position. Furthermore, this construction is conjectured to be optimal."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.15380","kind":"arxiv","version":3},"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/2307.15380/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"}