{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T03:38:29Z","timestamp":1772509109875,"version":"3.50.1"},"reference-count":22,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T00:00:00Z","timestamp":1557100800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T00:00:00Z","timestamp":1557100800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["678765"],"award-info":[{"award-number":["678765"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003977","name":"Israel Science Foundation","doi-asserted-by":"publisher","award":["1452\/15"],"award-info":[{"award-number":["1452\/15"]}],"id":[{"id":"10.13039\/501100003977","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006221","name":"United States-Israel Binational Science Foundation","doi-asserted-by":"publisher","award":["2014384"],"award-info":[{"award-number":["2014384"]}],"id":[{"id":"10.13039\/100006221","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":[[2020,6]]},"DOI":"10.1007\/s00454-019-00085-y","type":"journal-article","created":{"date-parts":[[2019,5,7]],"date-time":"2019-05-07T15:44:15Z","timestamp":1557243855000},"page":"848-866","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Further Consequences of the Colorful Helly Hypothesis"],"prefix":"10.1007","volume":"63","author":[{"given":"Leonardo","family":"Mart\u00ednez-Sandoval","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Edgardo","family":"Rold\u00e1n-Pensado","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Natan","family":"Rubin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,5,6]]},"reference":[{"issue":"3","key":"85_CR1","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1017\/S0963548300000225","volume":"1","author":"N Alon","year":"1992","unstructured":"Alon, N., B\u00e1r\u00e1ny, I., F\u00fcredi, Z., Kleitman, D.J.: Point selections and weak $$\\varepsilon $$-nets for convex hulls. Comb. Probab. Comput. 1(3), 189\u2013200 (1992)","journal-title":"Comb. Probab. Comput."},{"issue":"3\u20134","key":"85_CR2","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1007\/BF02574042","volume":"13","author":"N Alon","year":"1995","unstructured":"Alon, N., Kalai, G.: Bounding the piercing number. Discrete Comput. Geom. 13(3\u20134), 245\u2013256 (1995)","journal-title":"Discrete Comput. Geom."},{"issue":"1","key":"85_CR3","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/S0196-8858(02)00003-9","volume":"29","author":"N Alon","year":"2002","unstructured":"Alon, N., Kalai, G., Matou\u0161ek, J., Meshulam, R.: Transversal numbers for hypergraphs arising in geometry. Adv. Appl. Math. 29(1), 79\u2013101 (2002)","journal-title":"Adv. Appl. Math."},{"issue":"1","key":"85_CR4","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1016\/0001-8708(92)90052-M","volume":"96","author":"N Alon","year":"1992","unstructured":"Alon, N., Kleitman, D.J.: Piercing convex sets and the Hadwiger\u2013Debrunner $$(p, q)$$-problem. Adv. Math. 96(1), 103\u2013112 (1992)","journal-title":"Adv. Math."},{"key":"85_CR5","first-page":"55","volume-title":"Algebraic and Geometric Methods in Discrete Mathematics. Contemporary Mathematics","author":"N Amenta","year":"2017","unstructured":"Amenta, N., De Loera, J.A., Sober\u00f3n, P.: Helly\u2019s theorem: new variations and applications. In: Harrington, H.A., Omar, M., Wright, M. (eds.) Algebraic and Geometric Methods in Discrete Mathematics. Contemporary Mathematics, vol. 685, pp. 55\u201395. American Mathematical Society, Providence (2017)"},{"issue":"2","key":"85_CR6","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1007\/s00454-009-9180-4","volume":"42","author":"JL Arocha","year":"2009","unstructured":"Arocha, J.L., B\u00e1r\u00e1ny, I., Bracho, J., Fabila, R., Montejano, L.: Very colorful theorems. Discrete Comput. Geom. 42(2), 142\u2013154 (2009)","journal-title":"Discrete Comput. Geom."},{"issue":"2\u20133","key":"85_CR7","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1016\/0012-365X(82)90115-7","volume":"40","author":"I B\u00e1r\u00e1ny","year":"1982","unstructured":"B\u00e1r\u00e1ny, I.: A generalization of Carath\u00e9odory\u2019s theorem. Discrete Math. 40(2\u20133), 141\u2013152 (1982)","journal-title":"Discrete Math."},{"key":"85_CR8","first-page":"1","volume-title":"Geometry, Structure and Randomness in Combinatorics. Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series","author":"I B\u00e1r\u00e1ny","year":"2015","unstructured":"B\u00e1r\u00e1ny, I.: Tensors, colours, octahedra. In: Matou\u0161ek, J., Ne\u0161et\u0159il, J., Pellegrini, M. (eds.) Geometry, Structure and Randomness in Combinatorics. Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series, vol. 18, pp. 1\u201317. Edizioni della Normale, Pisa (2015)"},{"issue":"1","key":"85_CR9","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1007\/s11856-011-0029-1","volume":"182","author":"B Boris","year":"2011","unstructured":"Boris, B., Matou\u0161ek, J., Gabriel, N.: Lower bounds for weak epsilon-nets and stair-convexity. Isr. J. Math. 182(1), 199\u2013228 (2011). \n                  https:\/\/doi.org\/10.1007\/s11856-011-0029-1","journal-title":"Isr. J. Math."},{"issue":"1","key":"85_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF02574025","volume":"13","author":"B Chazelle","year":"1995","unstructured":"Chazelle, B., Edelsbrunner, H., Grigni, M., Guibas, L., Sharir, M., Welzl, E.: Improved bounds on weak $$\\epsilon $$-nets for convex sets. Discrete Comput. Geom. 13(1), 1\u201315 (1995). \n                  https:\/\/doi.org\/10.1007\/BF02574025","journal-title":"Discrete Comput. Geom."},{"issue":"5","key":"85_CR11","doi-asserted-by":"publisher","first-page":"347","DOI":"10.1007\/BF01900144","volume":"8","author":"L Danzer","year":"1957","unstructured":"Danzer, L.: \u00dcber ein Problem aus der kombinatorischen Geometrie. Arch. Math. (Basel) 8(5), 347\u2013351 (1957)","journal-title":"Arch. Math. (Basel)"},{"key":"85_CR12","doi-asserted-by":"crossref","unstructured":"Danzer, L., Gr\u00fcnbaum, B., Klee, V.: Helly\u2019s theorem and its relatives. In: 1963 Proceedings of the Symposium on Pure Mathematics, vol. VII, pp. 101\u2013180. American Mathematical Society, Providence (1963)","DOI":"10.1090\/pspum\/007\/0157289"},{"key":"85_CR13","doi-asserted-by":"crossref","unstructured":"Eckhoff, J.: Chapter: Helly, Radon, and Carath\u00e9odory type theorems. In: Handbook of Convex Geometry, pp. 389\u2013448. Elsevier, Amsterdam (1990)","DOI":"10.1016\/B978-0-444-89596-7.50017-1"},{"key":"85_CR14","doi-asserted-by":"crossref","unstructured":"Goodman, J.E., Pollack, R., Wenger, R.: Chapter: geometric transversal theory. In: New Trends in Discrete and Computational Geometry, pp. 163\u2013198. Springer, Berlin (1993)","DOI":"10.1007\/978-3-642-58043-7_8"},{"key":"85_CR15","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/BF01898794","volume":"8","author":"H Hadwiger","year":"1957","unstructured":"Hadwiger, H., Debrunner, H.: \u00dcber eine Variante zum Helly\u2019schen Satz. Arch. Math. (Basel) 8, 309\u2013313 (1957)","journal-title":"Arch. Math. (Basel)"},{"key":"85_CR16","first-page":"175","volume":"32","author":"E Helly","year":"1923","unstructured":"Helly, E.: \u00dcber Mengen konvexer K\u00f6rper mit gemeinschaftlichen Punkte. Jahresber. Dtsch. Math. Ver. 32, 175\u2013176 (1923)","journal-title":"Jahresber. Dtsch. Math. Ver."},{"issue":"6","key":"85_CR17","doi-asserted-by":"publisher","first-page":"633","DOI":"10.1007\/s00493-008-2427-5","volume":"28","author":"AF Holmsen","year":"2008","unstructured":"Holmsen, A.F., Pach, J., Tverberg, H.: Points surrounding the origin. Combinatorica 28(6), 633\u2013644 (2008)","journal-title":"Combinatorica"},{"issue":"2","key":"85_CR18","first-page":"284","volume":"75","author":"M Katchalski","year":"1979","unstructured":"Katchalski, M., Liu, A.: A problem of geometry in $$R^n$$. Proc. Am. Math. Soc. 75(2), 284\u2013288 (1979)","journal-title":"Proc. Am. Math. Soc."},{"key":"85_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-0039-7","volume-title":"Lectures on Discrete Geometry. Graduate Texts in Mathematics","author":"J Matou\u0161ek","year":"2002","unstructured":"Matou\u0161ek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer, New York (2002)"},{"key":"85_CR20","doi-asserted-by":"publisher","unstructured":"Rubin, N.: An improved bound for weak epsilon-nets in the plane. In: 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pp. 224\u2013235 (2018). \n                  https:\/\/doi.org\/10.1109\/FOCS.2018.00030","DOI":"10.1109\/FOCS.2018.00030"},{"issue":"4","key":"85_CR21","first-page":"49","volume":"II","author":"L Santal\u00f3","year":"1940","unstructured":"Santal\u00f3, L.: Un teorema sobre conjuntos de paralelep\u00edpedos de aristas paralelas. Publ. Inst. Mat. Univ. Nacl Litoral II(4), 49\u201360 (1940)","journal-title":"Publ. Inst. Mat. Univ. Nacl Litoral"},{"key":"85_CR22","unstructured":"Wenger, R., Holmsen, A.: Chapter: Helly-type theorems and geometric transversals. In: Handbook of Discrete and Computational Geometry, 3rd edn, pp. 91\u2013124. CRC Press LLC, Boca Raton (2017)"}],"container-title":["Discrete & Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-019-00085-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00454-019-00085-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-019-00085-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,9,16]],"date-time":"2020-09-16T10:17:04Z","timestamp":1600251424000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00454-019-00085-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,6]]},"references-count":22,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2020,6]]}},"alternative-id":["85"],"URL":"https:\/\/doi.org\/10.1007\/s00454-019-00085-y","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,5,6]]},"assertion":[{"value":"28 June 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 December 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 March 2019","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 May 2019","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}