{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T20:35:39Z","timestamp":1772742939875,"version":"3.50.1"},"reference-count":21,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2003,10,17]],"date-time":"2003-10-17T00:00:00Z","timestamp":1066348800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Random Struct Algorithms"],"published-print":{"date-parts":[[2003,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider a canonical Ramsey type problem. An edge\u2010coloring of a graph is called <jats:italic>m\u2010good<\/jats:italic> if each color appears at most <jats:italic>m<\/jats:italic> times at each vertex. Fixing a graph <jats:italic>G<\/jats:italic> and a positive integer <jats:italic>m<\/jats:italic>, let <jats:italic>f<\/jats:italic>(<jats:italic>m<\/jats:italic>, <jats:italic>G<\/jats:italic>) denote the smallest <jats:italic>n<\/jats:italic> such that every <jats:italic>m<\/jats:italic>\u2010good edge\u2010coloring of <jats:italic>K<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> yields a properly edge\u2010colored copy of <jats:italic>G<\/jats:italic>, and let <jats:italic>g<\/jats:italic>(<jats:italic>m<\/jats:italic>, <jats:italic>G<\/jats:italic>) denote the smallest <jats:italic>n<\/jats:italic> such that every <jats:italic>m<\/jats:italic>\u2010good edge\u2010coloring of <jats:italic>K<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> yields a rainbow copy of <jats:italic>G<\/jats:italic>. We give bounds on <jats:italic>f<\/jats:italic>(<jats:italic>m<\/jats:italic>, <jats:italic>G<\/jats:italic>) and <jats:italic>g<\/jats:italic>(<jats:italic>m<\/jats:italic>, <jats:italic>G<\/jats:italic>). For complete graphs <jats:italic>G = K<\/jats:italic><jats:sub><jats:italic>t<\/jats:italic><\/jats:sub>, we have <jats:italic>c<\/jats:italic><jats:sub>1<\/jats:sub><jats:italic>mt<\/jats:italic><jats:sup>2<\/jats:sup>\/ln <jats:italic>t \u2264 f<\/jats:italic>(<jats:italic>m<\/jats:italic>, <jats:italic>K<\/jats:italic><jats:sub><jats:italic>t<\/jats:italic><\/jats:sub>) \u2264 <jats:italic>c<\/jats:italic><jats:sub>2<\/jats:sub><jats:italic>mt<\/jats:italic><jats:sup>2<\/jats:sup>, and <jats:italic>c<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/tex2gif-stack-1.gif\" xlink:title=\"urn:x-wiley:10429832:media:RSA10102:tex2gif-stack-1\"\/><jats:italic>mt<\/jats:italic><jats:sup>3<\/jats:sup>\/ln <jats:italic>t \u2264 g<\/jats:italic>(<jats:italic>m<\/jats:italic>, <jats:italic>K<\/jats:italic><jats:sub><jats:italic>t<\/jats:italic><\/jats:sub>) \u2264 <jats:italic>c<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/tex2gif-stack-2.gif\" xlink:title=\"urn:x-wiley:10429832:media:RSA10102:tex2gif-stack-2\"\/><jats:italic>mt<\/jats:italic><jats:sup>3<\/jats:sup>\/ln <jats:italic>t<\/jats:italic>, where <jats:italic>c<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>c<\/jats:italic><jats:sub>2<\/jats:sub>, <jats:italic>c<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/tex2gif-stack-3.gif\" xlink:title=\"urn:x-wiley:10429832:media:RSA10102:tex2gif-stack-3\"\/>, <jats:italic>c<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/tex2gif-stack-4.gif\" xlink:title=\"urn:x-wiley:10429832:media:RSA10102:tex2gif-stack-4\"\/> are absolute constants. We also give bounds on <jats:italic>f<\/jats:italic>(<jats:italic>m<\/jats:italic>, <jats:italic>G<\/jats:italic>) and <jats:italic>g<\/jats:italic>(<jats:italic>m<\/jats:italic>, <jats:italic>G<\/jats:italic>) for general graphs <jats:italic>G<\/jats:italic> in terms of degrees in <jats:italic>G<\/jats:italic>. In particular, we show that for fixed <jats:italic>m<\/jats:italic> and <jats:italic>d<\/jats:italic>, and all sufficiently large <jats:italic>n<\/jats:italic> compared to <jats:italic>m<\/jats:italic> and <jats:italic>d<\/jats:italic>, <jats:italic>f<\/jats:italic>(<jats:italic>m<\/jats:italic>, <jats:italic>G<\/jats:italic>) = <jats:italic>n<\/jats:italic> for all graphs <jats:italic>G<\/jats:italic> with <jats:italic>n<\/jats:italic> vertices and maximum degree at most <jats:italic>d<\/jats:italic>. \u00a9 2003 Wiley Periodicals, Inc. Random Struct. Alg.,  2003<\/jats:p>","DOI":"10.1002\/rsa.10102","type":"journal-article","created":{"date-parts":[[2003,11,12]],"date-time":"2003-11-12T14:58:52Z","timestamp":1068649132000},"page":"409-433","source":"Crossref","is-referenced-by-count":48,"title":["Properly colored subgraphs and rainbow subgraphs in edge\u2010colorings with local constraints"],"prefix":"10.1002","volume":"23","author":[{"given":"Noga","family":"Alon","sequence":"first","affiliation":[]},{"given":"Tao","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Zevi","family":"Miller","sequence":"additional","affiliation":[]},{"given":"Dan","family":"Pritikin","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2003,10,17]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(82)90049-8"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240030102"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2418(199709)11:2<179::AID-RSA5>3.0.CO;2-P"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1002\/0471722154"},{"key":"e_1_2_1_6_2","first-page":"9","volume-title":"On an anti\u2010Ramsey type result, Colloq Math Soc Jan\u00f3s Bolyai 60, Sets, Graphs and Numbers","author":"Alon N.","year":"1991"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02582925"},{"key":"e_1_2_1_8_2","unstructured":"T.Bohman A.Frieze R.Martin M.Ruszink\u00f3 andC.Smyth Polychromatic cliques and related questions submitted."},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02756791"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(76)90053-8"},{"key":"e_1_2_1_11_2","unstructured":"G.Chen R.Schelp andB.Wei Monochromatic\u2010rainbow Ramsey numbers 14th Cumberland Conference Abstracts May2001 http:\/\/www.msci.memphis.edu\/balistep\/Abstracts.html."},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240060208"},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-25.4.249"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01788549"},{"key":"e_1_2_1_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(87)90017-7"},{"key":"e_1_2_1_16_2","article-title":"Constrained Ramsey numbers","author":"Jamison R.","journal-title":"J Graph Theory"},{"key":"e_1_2_1_17_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01294461"},{"key":"e_1_2_1_18_2","volume-title":"Graph colouring and the probabilistic method","author":"Molloy M.","year":"2001"},{"key":"e_1_2_1_19_2","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0118(199703)24:3<201::AID-JGT1>3.0.CO;2-T"},{"key":"e_1_2_1_20_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(79)90020-7"},{"key":"e_1_2_1_21_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01788530"},{"key":"e_1_2_1_22_2","volume-title":"Introduction to graph theory (2nd edition)","author":"West D. 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