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The new proof uses the recently introduced polynomial partitioning technique of Guth and Katz [12]. While this paper was still in a draft stage, a similar proof of our main result was posted to the arXiv by Joshua Zahl [28].<\/jats:p>","DOI":"10.1017\/s0963548312000144","type":"journal-article","created":{"date-parts":[[2012,4,25]],"date-time":"2012-04-25T08:27:14Z","timestamp":1335342434000},"page":"597-610","source":"Crossref","is-referenced-by-count":26,"title":["Unit Distances in Three Dimensions"],"prefix":"10.1017","volume":"21","author":[{"given":"HAIM","family":"KAPLAN","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"JI\u0158\u00cd","family":"MATOU\u0160EK","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ZUZANA","family":"SAFERNOV\u00c1","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"MICHA","family":"SHARIR","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2012,4,25]]},"reference":[{"key":"S0963548312000144_ref22","first-page":"293","volume-title":"Graph Theory and Combinatorics: Proc. 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First posted (v1) 26 April 2011; revised and corrected 22 September 2011."},{"key":"S0963548312000144_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-05355-3"},{"key":"S0963548312000144_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/BF01442458"},{"key":"S0963548312000144_ref10","first-page":"165","article-title":"On sets of distances on n points in Euclidean space.","volume":"5","author":"Erd\u0151s","year":"1960","journal-title":"Magyar Tud. Akad. Mat. Kutat\u00f3 Int. Kozl."},{"key":"S0963548312000144_ref9","doi-asserted-by":"publisher","DOI":"10.2307\/2305092"},{"key":"S0963548312000144_ref14","unstructured":"[14] Kaplan H. , Matou\u0161ek J. and Sharir M. (2011) Simple proofs of classical theorems in discrete geometry via the Guth\u2013Katz polynomial partitioning technique. Discrete Comput. Geom., submitted. Also in arXiv:1102.5391."},{"key":"S0963548312000144_ref20","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/342\/06151"},{"key":"S0963548312000144_ref3","doi-asserted-by":"publisher","DOI":"10.1112\/S0025579300011621"},{"key":"S0963548312000144_ref23","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-42-00925-6"},{"key":"S0963548312000144_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/BF02187783"},{"key":"S0963548312000144_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-009-9236-5"},{"key":"S0963548312000144_ref12","unstructured":"[12] Guth L. and Katz N. H. (2010) On the Erd\u0151s distinct distances problem in the plane. arXiv:1011.4105."},{"key":"S0963548312000144_ref6","volume-title":"Using Algebraic Geometry","author":"Cox","year":"2005"},{"key":"S0963548312000144_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-011-9391-3"},{"key":"S0963548312000144_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2010.11.008"},{"key":"S0963548312000144_ref24","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548397002976"},{"key":"S0963548312000144_ref25","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1515\/9781400874842-016","volume-title":"Differential and Combinatorial Topology","author":"Thom","year":"1965"},{"key":"S0963548312000144_ref18","first-page":"389","article-title":"On the topology of real algebraic surfaces.","volume":"13","author":"Oleinik","year":"1949","journal-title":"Izv. Akad. 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