{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:46:24Z","timestamp":1740109584974,"version":"3.37.3"},"reference-count":15,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,7,20]],"date-time":"2023-07-20T00:00:00Z","timestamp":1689811200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,7,20]],"date-time":"2023-07-20T00:00:00Z","timestamp":1689811200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100003825","name":"Magyar Tudom\u00e1nyos Akad\u00e9mia","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100003825","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":[[2023,10]]},"abstract":"Abstract<\/jats:title>In a colouring of $${\\mathbb {R}}^d$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> a pair $$(S,s_0)$$<\/jats:tex-math>\n \n (<\/mml:mo>\n S<\/mml:mi>\n ,<\/mml:mo>\n \n s<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with $$S\\subseteq {\\mathbb {R}}^d$$<\/jats:tex-math>\n \n S<\/mml:mi>\n \u2286<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and with $$s_0\\in S$$<\/jats:tex-math>\n \n \n s<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n \u2208<\/mml:mo>\n S<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is almost-monochromatic<\/jats:italic> if $$S\\setminus \\{s_0\\}$$<\/jats:tex-math>\n \n S<\/mml:mi>\n \\<\/mml:mo>\n {<\/mml:mo>\n \n s<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is monochromatic but S<\/jats:italic> is not. We consider questions about finding almost-monochromatic similar copies of pairs $$(S,s_0)$$<\/jats:tex-math>\n \n (<\/mml:mo>\n S<\/mml:mi>\n ,<\/mml:mo>\n \n s<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in colourings of\u00a0$${\\mathbb {R}}^d$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, $${\\mathbb {Z}}^d$$<\/jats:tex-math>\n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and of $${\\mathbb {Q}}$$<\/jats:tex-math>\n Q<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> under some restrictions on the colouring. Among other results, we characterise those $$(S,s_0)$$<\/jats:tex-math>\n \n (<\/mml:mo>\n S<\/mml:mi>\n ,<\/mml:mo>\n \n s<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with $$S\\subseteq {\\mathbb {Z}}$$<\/jats:tex-math>\n \n S<\/mml:mi>\n \u2286<\/mml:mo>\n Z<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for which every finite colouring of $${\\mathbb {R}}$$<\/jats:tex-math>\n R<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> without an infinite monochromatic arithmetic progression contains an almost-monochromatic similar copy of $$(S,s_0)$$<\/jats:tex-math>\n \n (<\/mml:mo>\n S<\/mml:mi>\n ,<\/mml:mo>\n \n s<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We also show that if $$S\\subseteq {\\mathbb {Z}}^d$$<\/jats:tex-math>\n \n S<\/mml:mi>\n \u2286<\/mml:mo>\n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$s_0$$<\/jats:tex-math>\n \n s<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is outside of the convex hull of $$S\\setminus \\{s_0\\}$$<\/jats:tex-math>\n \n S<\/mml:mi>\n \\<\/mml:mo>\n {<\/mml:mo>\n \n s<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, then every finite colouring of $${\\mathbb {R}}^d$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> without a monochromatic similar copy of $${\\mathbb {Z}}^d$$<\/jats:tex-math>\n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> contains an almost-monochromatic similar copy of $$(S,s_0)$$<\/jats:tex-math>\n \n (<\/mml:mo>\n S<\/mml:mi>\n ,<\/mml:mo>\n \n s<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Further, we propose an approach based on finding almost-monochromatic sets that might lead to a human-verifiable proof of $$\\chi ({{\\mathbb {R}}}^2)\\ge 5$$<\/jats:tex-math>\n \n \u03c7<\/mml:mi>\n (<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n \u2265<\/mml:mo>\n 5<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00454-023-00526-9","type":"journal-article","created":{"date-parts":[[2023,7,20]],"date-time":"2023-07-20T15:04:45Z","timestamp":1689865485000},"page":"753-772","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Almost-Monochromatic Sets and the Chromatic Number of the Plane"],"prefix":"10.1007","volume":"70","author":[{"given":"N\u00f3ra","family":"Frankl","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tam\u00e1s","family":"Hubai","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2970-0943","authenticated-orcid":false,"given":"D\u00f6m\u00f6t\u00f6r","family":"P\u00e1lv\u00f6lgyi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,7,20]]},"reference":[{"key":"526_CR1","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1016\/0097-3165(73)90011-3","volume":"14","author":"P Erd\u0151s","year":"1973","unstructured":"Erd\u0151s, P., Graham, R.L., Montgomery, P., Rothschild, B.L., Spencer, J., Straus, E.G.: Euclidean Ramsey theorems. 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Chapman and Hall, Boca Raton (2017)"},{"key":"526_CR6","unstructured":"Graham, R.L., Rothschild, B.L., Spencer, J.H.: Ramsey Theory. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1990)"},{"issue":"1","key":"526_CR8","first-page":"18","volume":"28","author":"ADNJ de Grey","year":"2018","unstructured":"de Grey, A.D.N.J.: The chromatic number of the plane is at least 5. Geombinatorics 28(1), 18\u201331 (2018)","journal-title":"Geombinatorics"},{"key":"526_CR7","unstructured":"de Grey, A.D.N.J.: Polymath proposal: finding simpler unit distance graphs of chromatic number\u00a05 (4\/10\/2018). https:\/\/polymathprojects.org\/2018\/04\/10\/polymath-proposal-finding-simpler-unit-distance-graphs-of-chromatic-number-5\/"},{"key":"526_CR9","doi-asserted-by":"publisher","first-page":"222","DOI":"10.1090\/S0002-9947-1963-0143712-1","volume":"106","author":"AW Hales","year":"1963","unstructured":"Hales, A.W., Jewett, R.I.: Regularity and positional games. 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