{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T16:56:37Z","timestamp":1782233797671,"version":"3.54.5"},"reference-count":17,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T00:00:00Z","timestamp":1761609600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T00:00:00Z","timestamp":1761609600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"publisher","award":["2347850"],"award-info":[{"award-number":["2347850"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2026,7]]},"DOI":"10.1007\/s00454-025-00761-2","type":"journal-article","created":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T15:26:23Z","timestamp":1761665183000},"page":"643-651","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Planar Point Sets with Forbidden 4-Point Patterns and Few Distinct Distances"],"prefix":"10.1007","volume":"76","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0140-7641","authenticated-orcid":false,"given":"Terence","family":"Tao","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,10,28]]},"reference":[{"key":"761_CR1","volume-title":"The Probabilistic Method","author":"N Alon","year":"2016","unstructured":"Alon, N., Spencer, J.: The Probabilistic Method, 4th edn. Wiley, New York (2016)","edition":"4"},{"key":"761_CR2","volume-title":"Research Problems in Discrete Geometry","author":"P Bra\u00df","year":"2005","unstructured":"Bra\u00df, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005)"},{"issue":"4","key":"761_CR3","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1016\/0020-0190(78)90067-4","volume":"7","author":"R DeMillo","year":"1978","unstructured":"DeMillo, R., Lipton, R.: A probabilistic remark on algebraic program testing. Inf. Process. Lett. 7(4), 193\u2013195 (1978)","journal-title":"Inf. Process. Lett."},{"issue":"2","key":"761_CR4","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1007\/s10998-008-8165-4","volume":"57","author":"A Dumitrescu","year":"2008","unstructured":"Dumitrescu, A.: On distinct distances among points in general position and other related problems. Period. Math. Hungar. 57(2), 165\u2013176 (2008)","journal-title":"Period. Math. Hungar."},{"issue":"9","key":"761_CR5","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2020.111967","volume":"343","author":"A Dumitrescu","year":"2020","unstructured":"Dumitrescu, A.: Distinct distances in planar point sets with forbidden 4-point patterns. Discret. Math. 343(9), 111967 (2020)","journal-title":"Discret. Math."},{"key":"761_CR6","doi-asserted-by":"publisher","first-page":"248","DOI":"10.1080\/00029890.1946.11991674","volume":"53","author":"P Erd\u0151s","year":"1946","unstructured":"Erd\u0151s, P.: On sets of distances of n points. Am. Math. Mon. 53, 248\u2013250 (1946)","journal-title":"Am. Math. Mon."},{"key":"761_CR7","doi-asserted-by":"crossref","unstructured":"Erd\u0151s, P.: On some problems of elementary and combinatorial geometry, Annali di Matematica Pura ed Applicata IV, Vol. CIII, pp. 99\u2013108 (1975)","DOI":"10.1007\/BF02414146"},{"key":"761_CR8","unstructured":"Erd\u0151s, P.: Extremal problems in number theory, combinatorics and geometry. In: Proceedings of the International Congress of Mathematicians August 16\u201324, Warszawa, pp. 51\u201370 (1983)"},{"key":"761_CR9","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1016\/0012-365X(86)90009-9","volume":"60","author":"P Erd\u0151s","year":"1986","unstructured":"Erd\u0151s, P.: On some metric and combinatorial geometric problems. Discrete Math. 60, 147\u2013153 (1986)","journal-title":"Discrete Math."},{"key":"761_CR10","unstructured":"Erd\u0151s, P.: Some old and new problems in combinatorial geometry. In: Applications of Discrete Mathematics (Clemson, SC, 1986). SIAM, Philadelphia, pp. 32\u201337 (1988)"},{"key":"761_CR11","first-page":"335","volume-title":"Graph Theory, Combinatorics, Algorithms and Applications","author":"P Erd\u0151s","year":"1995","unstructured":"Erd\u0151s, P.: Some of my recent problems in combinatorial number theory, geometry and combinatorics. In: Alavi, Y., et al. (eds.) Graph Theory, Combinatorics, Algorithms and Applications, vol. 1, pp. 335\u2013349. Wiley (1995)"},{"issue":"166","key":"761_CR12","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1016\/S0012-365X(96)00173-2","volume":"165","author":"P Erd\u0151s","year":"1997","unstructured":"Erd\u0151s, P.: Some old and new problems in various branches of combinatorics. Discrete Math. 165(166), 227\u2013231 (1997)","journal-title":"Discrete Math."},{"key":"761_CR13","doi-asserted-by":"publisher","first-page":"212","DOI":"10.1112\/jlms\/s1-16.4.212","volume":"16","author":"P Erd\u0151s","year":"1941","unstructured":"Erd\u0151s, P., Tur\u00e1n, P.: On a problem of Sidon in additive number theory, and on some related problems. J. Lond. Math. Soc. 16, 212\u2013215 (1941)","journal-title":"J. Lond. Math. Soc."},{"key":"761_CR14","doi-asserted-by":"publisher","first-page":"155","DOI":"10.4007\/annals.2015.181.1.2","volume":"181","author":"L Guth","year":"2015","unstructured":"Guth, L., Katz, N.: On the Erd\u0151s distinct distances problem in the plane. Ann. Math. 181, 155\u2013190 (2015)","journal-title":"Ann. Math."},{"issue":"4","key":"761_CR15","doi-asserted-by":"publisher","first-page":"701","DOI":"10.1145\/322217.322225","volume":"27","author":"JT Schwartz","year":"1980","unstructured":"Schwartz, J.T.: Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27(4), 701\u2013717 (1980)","journal-title":"J. ACM"},{"key":"761_CR16","doi-asserted-by":"publisher","first-page":"332","DOI":"10.1016\/0097-3165(95)90007-1","volume":"71","author":"T Thiele","year":"1995","unstructured":"Thiele, T.: The no-four-on-circle problem. J. Combin. Theory Ser. A 71, 332\u2013334 (1995)","journal-title":"J. Combin. Theory Ser. A"},{"key":"761_CR17","doi-asserted-by":"crossref","unstructured":"Zippel, R.: Probabilistic algorithms for sparse polynomials. In: Symbolic and Algebraic Computation, EUROSAM \u201979, An International Symposium on Symbolic and Algebraic Computation, Marseille, France, June 1979, Proceedings, Lecture Notes in Computer Science, Vol. 72, Springer, pp. 216\u2013226 (1979)","DOI":"10.1007\/3-540-09519-5_73"}],"container-title":["Discrete & Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-025-00761-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-025-00761-2","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-025-00761-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T16:41:41Z","timestamp":1782232901000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-025-00761-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,28]]},"references-count":17,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,7]]}},"alternative-id":["761"],"URL":"https:\/\/doi.org\/10.1007\/s00454-025-00761-2","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,28]]},"assertion":[{"value":"2 September 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 June 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 June 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 October 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}