{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T21:02:40Z","timestamp":1772744560138,"version":"3.50.1"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T00:00:00Z","timestamp":1696550400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T00:00:00Z","timestamp":1696550400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100011019","name":"Nemzeti Kutat\u00e1si Fejleszt\u00e9si \u00e9s Innov\u00e1ci\u00f3s Hivatal","doi-asserted-by":"publisher","award":["KKP-133819"],"award-info":[{"award-number":["KKP-133819"]}],"id":[{"id":"10.13039\/501100011019","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012550","name":"Nemzeti Kutat\u00e1si, Fejleszt\u00e9si \u00e9s Innovaci\u00f3s Alap","doi-asserted-by":"publisher","award":["TKP2021-NVA-09"],"award-info":[{"award-number":["TKP2021-NVA-09"]}],"id":[{"id":"10.13039\/501100012550","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012550","name":"Nemzeti Kutat\u00e1si, Fejleszt\u00e9si \u00e9s Innovaci\u00f3s Alap","doi-asserted-by":"publisher","award":["TKP2021-NKTA-62"],"award-info":[{"award-number":["TKP2021-NKTA-62"]}],"id":[{"id":"10.13039\/501100012550","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012550","name":"Nemzeti Kutat\u00e1si, Fejleszt\u00e9si \u00e9s Innovaci\u00f3s Alap","doi-asserted-by":"publisher","award":["132097"],"award-info":[{"award-number":["132097"]}],"id":[{"id":"10.13039\/501100012550","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012550","name":"Nemzeti Kutat\u00e1si, Fejleszt\u00e9si \u00e9s Innovaci\u00f3s Alap","doi-asserted-by":"publisher","award":["129335"],"award-info":[{"award-number":["129335"]}],"id":[{"id":"10.13039\/501100012550","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","award":["Geoscape"],"award-info":[{"award-number":["Geoscape"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Program."],"published-print":{"date-parts":[[2024,9]]},"DOI":"10.1007\/s10107-023-02012-9","type":"journal-article","created":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T15:01:41Z","timestamp":1696604501000},"page":"303-327","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["The density of planar sets avoiding unit distances"],"prefix":"10.1007","volume":"207","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1246-6601","authenticated-orcid":false,"given":"Gergely","family":"Ambrus","sequence":"first","affiliation":[]},{"given":"Adri\u00e1n","family":"Csisz\u00e1rik","sequence":"additional","affiliation":[]},{"given":"M\u00e1t\u00e9","family":"Matolcsi","sequence":"additional","affiliation":[]},{"given":"D\u00e1niel","family":"Varga","sequence":"additional","affiliation":[]},{"given":"P\u00e1l","family":"Zs\u00e1mboki","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,10,6]]},"reference":[{"key":"2012_CR1","doi-asserted-by":"publisher","first-page":"1245","DOI":"10.1007\/s00454-020-00263-3","volume":"67","author":"G Ambrus","year":"2022","unstructured":"Ambrus, G., Matolcsi, M.: Density estimates of 1-avoiding sets via higher order correlations. Discrete Comput. Geom. 67, 1245\u20131256 (2022)","journal-title":"Discrete Comput. Geom."},{"key":"2012_CR2","doi-asserted-by":"publisher","first-page":"783","DOI":"10.1007\/s00454-015-9668-z","volume":"53","author":"C Bachoc","year":"2015","unstructured":"Bachoc, C., Passuello, A., Thiery, A.: The density of sets avoiding distance 1 in Euclidean space. Discrete Comput. Geom. 53, 783\u2013808 (2015)","journal-title":"Discrete Comput. Geom."},{"issue":"1","key":"2012_CR3","doi-asserted-by":"publisher","first-page":"5153","DOI":"10.46298\/dmtcs.5153","volume":"23","author":"T Bellitto","year":"2021","unstructured":"Bellitto, T., P\u00eacher, A., S\u00e9dillot, A.: On the density of sets of the Euclidean plane avoiding distance 1. Discrete Math. Theor. Comput. Sci. 23(1), 5153 (2021)","journal-title":"Discrete Math. Theor. Comput. Sci."},{"key":"2012_CR4","unstructured":"Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer (2005)"},{"key":"2012_CR5","doi-asserted-by":"publisher","first-page":"837","DOI":"10.1007\/s00493-016-3380-3","volume":"37","author":"DW Cranston","year":"2017","unstructured":"Cranston, D.W., Rabern, L.: The fractional chromatic number of the plane. Combinatorica 37, 837\u2013861 (2017)","journal-title":"Combinatorica"},{"key":"2012_CR6","first-page":"22","volume":"30","author":"HT Croft","year":"1967","unstructured":"Croft, H.T.: Incidence incidents. Eureka 30, 22\u201326 (1967)","journal-title":"Eureka"},{"key":"2012_CR7","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1007\/s10107-020-01562-6","volume":"191","author":"E DeCorte","year":"2022","unstructured":"DeCorte, E., de Oliveira Filho, F.M., Vallentin, F.: Complete positivity and distance-avoiding sets. Math. Program. 191, 487\u2013558 (2022)","journal-title":"Math. Program."},{"issue":"1","key":"2012_CR8","first-page":"18","volume":"28","author":"A de Grey","year":"2018","unstructured":"de Grey, A.: The chromatic number of the plane is at least 5. Geombinatorics 28(1), 18\u201331 (2018)","journal-title":"Geombinatorics"},{"key":"2012_CR9","doi-asserted-by":"publisher","first-page":"1417","DOI":"10.4171\/jems\/236","volume":"12","author":"FM de Oliveira Filho","year":"2010","unstructured":"de Oliveira Filho, F.M., Vallentin, F.: Fourier analysis, linear programming, and densities of distance-avoiding sets in $$\\mathbb{R} ^n$$. J. Eur. Math. Soc. 12, 1417\u20131428 (2010)","journal-title":"J. Eur. Math. Soc."},{"issue":"3","key":"2012_CR10","doi-asserted-by":"publisher","first-page":"785","DOI":"10.1112\/S0025579319000160","volume":"65","author":"FM de Oliveira Filho","year":"2019","unstructured":"de Oliveira Filho, F.M., Vallentin, F.M.: A counterexample to a conjecture of Larman and Rogers on sets avoiding distance 1. Mathematika 65(3), 785\u2013787 (2019)","journal-title":"Mathematika"},{"key":"2012_CR11","doi-asserted-by":"crossref","unstructured":"Erd\u0151s, P.: Problems and results in combinatorial geometry. In: Discrete Geometry and Convexity (New York, 1982), Annals of the New York Academy of Sciences, vol. 440, pp. 1\u201311. New York Academy of Sciences, New York (1985)","DOI":"10.1111\/j.1749-6632.1985.tb14533.x"},{"key":"2012_CR12","doi-asserted-by":"publisher","first-page":"216","DOI":"10.1007\/s00454-019-00058-1","volume":"64","author":"G Exoo","year":"2020","unstructured":"Exoo, G., Ismailescu, D.: The chromatic number of the plane is at least 5: a new proof. Discrete Comput. Geom. 64, 216\u2013226 (2020)","journal-title":"Discrete Comput. Geom."},{"issue":"2","key":"2012_CR13","doi-asserted-by":"publisher","first-page":"184","DOI":"10.1016\/0097-3165(81)90014-5","volume":"31","author":"KJ Falconer","year":"1981","unstructured":"Falconer, K.J.: The realization of distances in measurable subsets covering $$\\mathbb{R}^n$$. J. Comb. Theory Ser. A 31(2), 184\u2013189 (1981)","journal-title":"J. Comb. Theory Ser. A"},{"key":"2012_CR14","first-page":"8","volume":"2","author":"D Fisher","year":"1992","unstructured":"Fisher, D., Ullman, D.: The fractional chromatic number of the plane. Geombinatorics 2, 8\u201312 (1992)","journal-title":"Geombinatorics"},{"key":"2012_CR15","unstructured":"Gurobi Optimization, LLC, Gurobi Optimizer Reference Manual. https:\/\/www.gurobi.com (2022)"},{"key":"2012_CR16","first-page":"238","volume":"4","author":"H Hadwiger","year":"1945","unstructured":"Hadwiger, H.: \u00dcberdeckung des Euklidischen Raumes durch kongruente Mengen. Port. Math. 4, 238\u2013242 (1945)","journal-title":"Port. Math."},{"issue":"7825","key":"2012_CR17","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1038\/s41586-020-2649-2","volume":"585","author":"CR Harris","year":"2020","unstructured":"Harris, C.R., Millman, K.J., Van Der Walt, S.J., Gommers, R., Virtanen, P., Cournapeau, D., Wieser, E., Taylor, J., Berg, S., Smith, N.J., Kern, R.: Array programming with NumPy. Nature 585(7825), 357\u2013362 (2020)","journal-title":"Nature"},{"key":"2012_CR18","unstructured":"https:\/\/arxiv.org\/abs\/2207.14179"},{"key":"2012_CR19","unstructured":"https:\/\/bit.ly\/unit-distances"},{"key":"2012_CR20","volume-title":"An Introduction to Harmonic Analysis","author":"Y Katznelson","year":"1968","unstructured":"Katznelson, Y.: An Introduction to Harmonic Analysis. Wiley, New York (1968)"},{"key":"2012_CR21","doi-asserted-by":"publisher","first-page":"642","DOI":"10.1007\/s00454-015-9751-5","volume":"55","author":"T Keleti","year":"2016","unstructured":"Keleti, T., Matolcsi, M., de Oliveira Filho, F.M., Ruzsa, I.Z.: Better bounds for planar sets avoiding unit distances. Discrete Comput. Geom. 55, 642\u2013661 (2016)","journal-title":"Discrete Comput. Geom."},{"key":"2012_CR22","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1112\/S0025579300004903","volume":"19","author":"DG Larman","year":"1972","unstructured":"Larman, D.G., Rogers, C.A.: The realization of distances within sets in Euclidean space. Mathematika 19, 1\u201324 (1972)","journal-title":"Mathematika"},{"key":"2012_CR23","doi-asserted-by":"publisher","first-page":"2376","DOI":"10.7717\/peerj-cs.103","volume":"3","author":"A Meurer","year":"2017","unstructured":"Meurer, A., Smith, C.P., Paprocki, M., Certik, O., Kirpichev, S.B., Rocklin, M., Kumar, A., Ivanov, S., Moore, J.K., Singh, S., Rathnayake, T.: SymPy: symbolic computing in Python. PeerJ Comput. Sci. 3, 2376\u20135992 (2017)","journal-title":"PeerJ Comput. Sci."},{"key":"2012_CR24","unstructured":"MOSEK ApS, MOSEK Optimizer API for Python 9.3.20. https:\/\/docs.mosek.com\/latest\/pythonapi\/index.html (2022)"},{"key":"2012_CR25","unstructured":"Moser, L.: Poorly formulated unsolved problems of combinatorial geometry. Mimeographed (1966)"},{"key":"2012_CR26","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1017\/S0008439500025753","volume":"4","author":"L Moser","year":"1961","unstructured":"Moser, L., Moser, W.: Solution to problem 10. Can. Math. Bull. 4, 187\u2013189 (1961)","journal-title":"Can. Math. Bull."},{"key":"2012_CR27","unstructured":"Parts, J.: PolyMath16, Comment 37575. https:\/\/dustingmixon.wordpress.com\/2021\/02\/01\/polymath16-seventeenth-thread-declaring-victory\/#comment-37575"},{"key":"2012_CR28","unstructured":"PolyMath16: Hadwiger-Nelson problem. https:\/\/asone.ai\/polymath\/index.php?title=Hadwiger-Nelson_problem (2022)"},{"key":"2012_CR29","unstructured":"Scheinerman, E.R., Ullman, D.H.: Fractional Graph Theory: A Rational Approach to the Theory of Graphs. Courier Corporation (2011)"},{"key":"2012_CR30","doi-asserted-by":"publisher","first-page":"811","DOI":"10.2307\/1968466","volume":"39","author":"IJ Schoenberg","year":"1938","unstructured":"Schoenberg, I.J.: Metric spaces and completely monotone functions. Ann. Math. 39, 811\u2013841 (1938)","journal-title":"Ann. Math."},{"key":"2012_CR31","doi-asserted-by":"crossref","unstructured":"Soifer, A.: The mathematical coloring book. Mathematics of coloring and the colorful life of its creators. With forewords by Branko Gr\u00fcnbaum, Peter D. Johnson, Jr. and Cecil Rousseau. Springer, New York (2009)","DOI":"10.1007\/978-0-387-74642-5"},{"key":"2012_CR32","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1007\/BF02579223","volume":"4","author":"LA Sz\u00e9kely","year":"1984","unstructured":"Sz\u00e9kely, L.A.: Measurable chromatic number of geometric graphs and sets without some distances in Euclidean space. Combinatorica 4, 213\u2013218 (1984)","journal-title":"Combinatorica"},{"key":"2012_CR33","first-page":"646","volume-title":"Paul Erd\u0151s and His Mathematics. II. Bolyai Society Mathematical Studies","author":"LA Sz\u00e9kely","year":"2002","unstructured":"Sz\u00e9kely, L.A.: Erd\u0151s on unit distances and the Szemer\u00e9di\u2013Trotter theorems. In: Hal\u00e1sz, G., Lov\u00e1sz, L., Simonovits, M., S\u00f3s, V.T. (eds.) Paul Erd\u0151s and His Mathematics. II. Bolyai Society Mathematical Studies, vol. 11, pp. 646\u2013666. J\u00e1nos Bolyai Mathematical Society, Springer-Verlag, Berlin (2002)"},{"key":"2012_CR34","volume-title":"A Treatise on the Theory of Bessel Functions","author":"GN Watson","year":"1922","unstructured":"Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1922)"}],"container-title":["Mathematical Programming"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-023-02012-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10107-023-02012-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-023-02012-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,7]],"date-time":"2024-08-07T15:07:37Z","timestamp":1723043257000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10107-023-02012-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,6]]},"references-count":34,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2024,9]]}},"alternative-id":["2012"],"URL":"https:\/\/doi.org\/10.1007\/s10107-023-02012-9","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"value":"0025-5610","type":"print"},{"value":"1436-4646","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,6]]},"assertion":[{"value":"11 November 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 July 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 October 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}