{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,23]],"date-time":"2023-10-23T06:40:40Z","timestamp":1698043240929},"reference-count":4,"publisher":"Wiley","issue":"5","license":[{"start":{"date-parts":[[2006,10,6]],"date-time":"2006-10-06T00:00:00Z","timestamp":1160092800000},"content-version":"vor","delay-in-days":5818,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1990,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In the graph <jats:italic>K<jats:sub>m<\/jats:sub><\/jats:italic> \u00d7 <jats:italic>K<jats:sub>m<\/jats:sub><\/jats:italic> a 4\u2010cycle is called square if its vertices do not all lie in the same copy of <jats:italic>K<jats:sub>m<\/jats:sub><\/jats:italic>. A <jats:italic>c<\/jats:italic>\u2010coloring of the edges of <jats:italic>K<jats:sub>m<\/jats:sub><\/jats:italic> \u00d7 <jats:italic>K<jats:sub>m<\/jats:sub><\/jats:italic> is good if in any square 4\u2010cycle the two edges of at least one nonadjacent pair receive different colors. Let <jats:italic>f<\/jats:italic>(2,<jats:italic>c<\/jats:italic>) denote the smallest integer <jats:italic>m<\/jats:italic> so that no <jats:italic>c<\/jats:italic>\u2010coloring of <jats:italic>K<jats:sub>m<\/jats:sub><\/jats:italic> \u00d7 <jats:italic>K<jats:sub>m<\/jats:sub><\/jats:italic> is good. We show that there exists a constant <jats:italic>k<\/jats:italic> so that <jats:italic>f<\/jats:italic>(2,<jats:italic>c<\/jats:italic>) &gt; <jats:italic>kc<\/jats:italic><jats:sup>3<\/jats:sup>. (This function was introduced by Shelah in his proof of van der Waerden's theorem).<\/jats:p>","DOI":"10.1002\/jgt.3190140509","type":"journal-article","created":{"date-parts":[[2007,5,26]],"date-time":"2007-05-26T12:50:37Z","timestamp":1180183837000},"page":"575-583","source":"Crossref","is-referenced-by-count":3,"title":["Coloring the edges of <i>k<sub>m<\/sub><\/i> \u00d7 <i>k<sub>m<\/sub><\/i>"],"prefix":"10.1002","volume":"14","author":[{"given":"Katherine","family":"Heinrich","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,6]]},"reference":[{"key":"e_1_2_1_2_2","volume-title":"Ramsey Theory","author":"Graham R. L.","year":"1980"},{"key":"e_1_2_1_3_2","unstructured":"Saharon Shelah Ronald Graham's triumph: how the van der Waerden numbers fell from the Ackerman sphere into the atmosphere. Preprint."},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190070104"},{"key":"e_1_2_1_5_2","first-page":"212","article-title":"Beweis einer Baudetschen Vermutung","volume":"15","author":"van der Waerden B. L.","year":"1927","journal-title":"Nieuw Archief voor Wiskunde"}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190140509","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190140509","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,22]],"date-time":"2023-10-22T19:06:13Z","timestamp":1698001573000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190140509"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,11]]},"references-count":4,"journal-issue":{"issue":"5","published-print":{"date-parts":[[1990,11]]}},"alternative-id":["10.1002\/jgt.3190140509"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190140509","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,11]]}}}