{"paper":{"title":"On triple intersections of three families of unit circles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"J\\'ozsef Solymosi, Micha Sharir, Orit E. Raz","submitted_at":"2014-07-24T15:50:03Z","abstract_excerpt":"Let $p_1,p_2,p_3$ be three distinct points in the plane, and, for $i=1,2,3$, let $\\mathcal C_i$ be a family of $n$ unit circles that pass through $p_i$. We address a conjecture made by Sz\\'ekely, and show that the number of points incident to a circle of each family is $O(n^{11/6})$, improving an earlier bound for this problem due to Elekes, Simonovits, and Szab\\'o [Combin. Probab. Comput., 2009]. The problem is a special instance of a more general problem studied by Elekes and Szab\\'o [Combinatorica, 2012] (and by Elekes and R\\'onyai [J. Combin. Theory Ser. A, 2000])."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6625","kind":"arxiv","version":2},"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"}