{"paper":{"title":"A Crossing Lemma for Jordan Curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"G\\'abor Tardos, J\\'anos Pach, Natan Rubin","submitted_at":"2017-08-07T11:40:29Z","abstract_excerpt":"If two Jordan curves in the plane have precisely one point in common, and there they do not properly cross, then the common point is called a {\\em touching point}. The main result of this paper is a Crossing Lemma for simple curves: Let $X$ and $T$ stand for the sets of intersection points and touching points, respectively, in a family of $n$ simple curves in the plane, no three of which pass through the same point. If $|T|>cn$, for some fixed constant $c>0$, then we prove that $|X|=\\Omega(|T|(\\log\\log(|T|/n))^{1/504})$. In particular, if $|T|/n\\rightarrow\\infty$, then the number of intersecti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.02077","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":""},"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"}