{"paper":{"title":"A sharp threshold phenomenon in string graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Istvan Tomon","submitted_at":"2019-08-15T14:02:54Z","abstract_excerpt":"We prove that for every $\\epsilon>0$ there exists $\\delta>0$ such that the following holds. Let $\\mathcal{C}$ be a collection of $n$ curves in the plane such that there are at most $(\\frac{1}{4}-\\epsilon)\\frac{n^{2}}{2}$ pairs of curves $\\{\\alpha,\\beta\\}$ in $\\mathcal{C}$ having a nonempty intersection. Then $\\mathcal{C}$ contains two disjoint subsets $\\mathcal{A}$ and $\\mathcal{B}$ such that $|\\mathcal{A}|=|\\mathcal{B}|\\geq \\delta n$, and every $\\alpha\\in \\mathcal{A}$ is disjoint from every $\\beta\\in\\mathcal{B}$. On the other hand, for every positive integer $n$ there exists a collection $\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.05550","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/1908.05550/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"}