{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,12]],"date-time":"2026-01-12T11:49:23Z","timestamp":1768218563085,"version":"3.49.0"},"reference-count":27,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T00:00:00Z","timestamp":1762128000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2026,1]]},"abstract":"Abstract<\/jats:title>\n \n A finite point set in\n \n \n \n $\\mathbb{R}^d$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is in general position if no\n \n \n \n $d + 1$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n points lie on a common hyperplane. Let\n \n \n \n $\\alpha _d(N)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n be the largest integer such that any set of\n \n \n \n $N$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n points in\n \n \n \n $\\mathbb{R}^d$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , with no\n \n \n \n $d + 2$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n members on a common hyperplane, contains a subset of size\n \n \n \n $\\alpha _d(N)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in general position. Using the method of hypergraph containers, Balogh and Solymosi showed that\n \n \n \n $\\alpha _2(N) \\lt N^{5\/6 + o(1)}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . In this paper, we also use the container method to obtain new upper bounds for\n \n \n \n $\\alpha _d(N)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n when\n \n \n \n $d \\geq 3$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . More precisely, we show that if\n \n \n \n $d$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is odd, then\n \n \n \n $\\alpha _d(N) \\lt N^{\\frac {1}{2} + \\frac {1}{2d} + o(1)}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , and if\n \n \n \n $d$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is even, we have\n \n \n \n $\\alpha _d(N) \\lt N^{\\frac {1}{2} + \\frac {1}{d-1} + o(1)}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . We also study the classical problem of determining\n \n \n \n $a(d,k,n)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , the maximum number of points selected from the grid\n \n \n \n $[n]^d$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n such that no\n \n \n \n $k + 2$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n members lie on a\n \n \n \n $k$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -flat, and improve the previously best known bound for\n \n \n \n $a(d,k,n)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , due to Lefmann in 2008, by a polynomial factor when\n \n \n \n $k$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n = 2 or 3 (mod 4).\n <\/jats:p>","DOI":"10.1017\/s0963548325100254","type":"journal-article","created":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T08:14:30Z","timestamp":1762157670000},"page":"134-148","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["On higher dimensional point sets in general position"],"prefix":"10.1017","volume":"35","author":[{"given":"Andrew","family":"Suk","sequence":"first","affiliation":[{"name":"University of California at San Diego"}]},{"given":"Ji","family":"Zeng","sequence":"additional","affiliation":[{"name":"University of California at San Diego"}]}],"member":"56","published-online":{"date-parts":[[2025,11,3]]},"reference":[{"key":"S0963548325100254_ref8","doi-asserted-by":"publisher","DOI":"10.1145\/2213977.2214010"},{"key":"S0963548325100254_ref10","doi-asserted-by":"publisher","DOI":"10.1006\/jcta.1997.2829"},{"key":"S0963548325100254_ref4","doi-asserted-by":"publisher","DOI":"10.1016\/S0925-7721(02)00127-X"},{"key":"S0963548325100254_ref18","unstructured":"[18] Lefmann, H. (2012) Extensions of the no-three-in-line problem, Preprint, http:\/\/www.tu-chemnitz.de\/informatik\/ThIS\/downloads\/publications\/lefmann_no_three_submitted.pdf."},{"key":"S0963548325100254_ref11","doi-asserted-by":"publisher","DOI":"10.1137\/0404019"},{"key":"S0963548325100254_ref12","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(89)90089-7"},{"key":"S0963548325100254_ref15","doi-asserted-by":"publisher","DOI":"10.1006\/jnth.1993.1037"},{"key":"S0963548325100254_ref1","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-2014-00816-X"},{"key":"S0963548325100254_ref14","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(75)90043-6"},{"key":"S0963548325100254_ref6","first-page":"P4","article-title":"$k$\n\n\n-fold Sidon sets","volume":"21","author":"Cilleruelo","year":"2014","journal-title":"Electron. J. Comb."},{"key":"S0963548325100254_ref22","first-page":"167","article-title":"Steiner triple systems with minimum independence number","volume":"21","author":"Phelps","year":"1986","journal-title":"ARS Combin."},{"key":"S0963548325100254_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(86)90009-9"},{"key":"S0963548325100254_ref3","first-page":"20","article-title":"On the number of points in general position in the plane","volume":"16","author":"Balogh","year":"2018","journal-title":"Discrete Anal."},{"key":"S0963548325100254_ref24","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-26.3.198"},{"key":"S0963548325100254_ref27","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-023-00601-1"},{"key":"S0963548325100254_ref2","doi-asserted-by":"crossref","unstructured":"[2] Balogh, J. , Morris, R. and Samotij, W. (2018) The method of hypergraph containers. In: Proceedings of the International Congress of Mathematicians: Rio de Janeiro 2018. World Scientific, pp. 3059\u20133092.","DOI":"10.1142\/9789813272880_0172"},{"key":"S0963548325100254_ref19","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548317000098"},{"key":"S0963548325100254_ref16","doi-asserted-by":"publisher","DOI":"10.37236\/1718"},{"key":"S0963548325100254_ref17","doi-asserted-by":"crossref","unstructured":"[17] Lefmann, H. (2008) No $\\ell$ grid-points in spaces of small dimension. In Algorithmic Aspects in Information and Management: 4th International Conference, AAIM. Springer, pp. 259\u2013270.","DOI":"10.1007\/978-3-540-68880-8_25"},{"key":"S0963548325100254_ref26","doi-asserted-by":"publisher","DOI":"10.1007\/s00222-014-0562-8"},{"key":"S0963548325100254_ref23","doi-asserted-by":"publisher","DOI":"10.1007\/s00453-006-0158-9"},{"key":"S0963548325100254_ref21","doi-asserted-by":"publisher","DOI":"10.37236\/32"},{"key":"S0963548325100254_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/BF03041066"},{"key":"S0963548325100254_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/s00022-016-0323-5"},{"key":"S0963548325100254_ref20","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2016.05.001"},{"key":"S0963548325100254_ref25","doi-asserted-by":"publisher","DOI":"10.4064\/aa-65-3-259-282"},{"key":"S0963548325100254_ref7","unstructured":"[7] Dudeney, H. E. (1917) Amusements in Mathematics. Nelson, London."}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548325100254","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,12]],"date-time":"2026-01-12T08:49:26Z","timestamp":1768207766000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548325100254\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,3]]},"references-count":27,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["S0963548325100254"],"URL":"https:\/\/doi.org\/10.1017\/s0963548325100254","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,3]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}