{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,15]],"date-time":"2024-09-15T18:47:01Z","timestamp":1726426021236},"reference-count":29,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2020,3,9]],"date-time":"2020-03-09T00:00:00Z","timestamp":1583712000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2020,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>For an integer<jats:italic>q<\/jats:italic>\u2a7e 2, a graph<jats:italic>G<\/jats:italic>is called<jats:italic>q<\/jats:italic>-Ramsey for a graph<jats:italic>H<\/jats:italic>if every<jats:italic>q<\/jats:italic>-colouring of the edges of<jats:italic>G<\/jats:italic>contains a monochromatic copy of<jats:italic>H<\/jats:italic>. If<jats:italic>G<\/jats:italic>is<jats:italic>q<\/jats:italic>-Ramsey for<jats:italic>H<\/jats:italic>yet no proper subgraph of<jats:italic>G<\/jats:italic>has this property, then<jats:italic>G<\/jats:italic>is called<jats:italic>q<\/jats:italic>-Ramsey-minimal for<jats:italic>H<\/jats:italic>. Generalizing a statement by Burr, Ne\u0161et\u0159il and R\u00f6dl from 1977, we prove that, for<jats:italic>q<\/jats:italic>\u2a7e 3, if<jats:italic>G<\/jats:italic>is a graph that is not<jats:italic>q<\/jats:italic>-Ramsey for some graph<jats:italic>H<\/jats:italic>, then<jats:italic>G<\/jats:italic>is contained as an induced subgraph in an infinite number of<jats:italic>q<\/jats:italic>-Ramsey-minimal graphs for<jats:italic>H<\/jats:italic>as long as<jats:italic>H<\/jats:italic>is 3-connected or isomorphic to the triangle. For such<jats:italic>H<\/jats:italic>, the following are some consequences.<\/jats:p><jats:p><jats:list list-type=\"bullet\"><jats:list-item><jats:p>For 2 \u2a7d<jats:italic>r<\/jats:italic>&lt;<jats:italic>q<\/jats:italic>, every<jats:italic>r<\/jats:italic>-Ramsey-minimal graph for<jats:italic>H<\/jats:italic>is contained as an induced subgraph in an infinite number of<jats:italic>q<\/jats:italic>-Ramsey-minimal graphs for<jats:italic>H<\/jats:italic>.<\/jats:p><\/jats:list-item><jats:list-item><jats:p>For every<jats:italic>q<\/jats:italic>\u2a7e 3, there are<jats:italic>q<\/jats:italic>-Ramsey-minimal graphs for<jats:italic>H<\/jats:italic>of arbitrarily large maximum degree, genus and chromatic number.<\/jats:p><\/jats:list-item><jats:list-item><jats:p>The collection<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548320000036_inline1.png\"\/><jats:tex-math>$\\{\\mathcal M_q(H) \\colon H \\text{ is 3-connected or } K_3\\}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>forms an antichain with respect to the subset relation, where<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548320000036_inline2.png\"\/><jats:tex-math>$\\mathcal M_q(H)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>denotes the set of all graphs that are<jats:italic>q<\/jats:italic>-Ramsey-minimal for<jats:italic>H<\/jats:italic>.<\/jats:p><\/jats:list-item><\/jats:list><\/jats:p><jats:p>We also address the question of which pairs of graphs satisfy<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548320000036_inline3.png\"\/><jats:tex-math>$\\mathcal M_q(H_1)=\\mathcal M_q(H_2)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, in which case<jats:italic>H<\/jats:italic><jats:sub>1<\/jats:sub>and<jats:italic>H<\/jats:italic><jats:sub>2<\/jats:sub>are called<jats:italic>q<\/jats:italic>-equivalent. We show that two graphs<jats:italic>H<\/jats:italic><jats:sub>1<\/jats:sub>and<jats:italic>H<\/jats:italic><jats:sub>2<\/jats:sub>are<jats:italic>q<\/jats:italic>-equivalent for even<jats:italic>q<\/jats:italic>if they are 2-equivalent, and that in general<jats:italic>q<\/jats:italic>-equivalence for some<jats:italic>q<\/jats:italic>\u2a7e 3 does not necessarily imply 2-equivalence. Finally we indicate that for connected graphs this implication may hold: results by Ne\u0161et\u0159il and R\u00f6dl and by Fox, Grinshpun, Liebenau, Person and Szab\u00f3 imply that the complete graph is not 2-equivalent to any other connected graph. We prove that this is the case for an arbitrary number of colours.<\/jats:p>","DOI":"10.1017\/s0963548320000036","type":"journal-article","created":{"date-parts":[[2020,3,9]],"date-time":"2020-03-09T07:24:45Z","timestamp":1583738685000},"page":"537-554","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":3,"title":["On minimal Ramsey graphs and Ramsey equivalence in multiple colours"],"prefix":"10.1017","volume":"29","author":[{"given":"Dennis","family":"Clemens","sequence":"first","affiliation":[]},{"given":"Anita","family":"Liebenau","sequence":"additional","affiliation":[]},{"given":"Damian","family":"Reding","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2020,3,9]]},"reference":[{"key":"S0963548320000036_ref17","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.20199"},{"key":"S0963548320000036_ref15","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.jctb.2014.06.003","article-title":"What is Ramsey-equivalent to a clique?","volume":"109","author":"Fox","year":"2014","journal-title":"J. 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In Proc. 8th Southeastern Conference on Combinatorics, Graph Theory and Computing, pp. 115\u2013124."},{"key":"S0963548320000036_ref25","doi-asserted-by":"publisher","DOI":"10.2307\/2152833"},{"key":"S0963548320000036_ref24","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s2-30.1.264"},{"key":"S0963548320000036_ref1","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1002\/jgt.22119","article-title":"Conditions on Ramsey nonequivalence","volume":"86","author":"Axenovich","year":"2017","journal-title":"J. Graph Theory"},{"key":"S0963548320000036_ref2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.20118"},{"key":"S0963548320000036_ref3","doi-asserted-by":"crossref","first-page":"P3.4","DOI":"10.37236\/7554","article-title":"Ramsey equivalence of Kn and Kn + Kn-1","volume":"25","author":"Bloom","year":"2018","journal-title":"Electron. J. Combin"},{"key":"S0963548320000036_ref4","unstructured":"[4] Bodkin, C. (2016) Folkman numbers and a conjecture in Ramsey theory. 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