{"title":"Chromatic Number","authors":"","doi":"10.1017/9781108912181.012","DOIUrl":"https://doi.org/10.1017/9781108912181.012","url":null,"abstract":"","PeriodicalId":179047,"journal":{"name":"The Discrete Mathematical Charms of Paul Erdős","volume":"92 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115669259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Van Der Waerden’s Theorem","authors":"Ted Alper","doi":"10.1090/cbms/123/04","DOIUrl":"https://doi.org/10.1090/cbms/123/04","url":null,"abstract":"Let us show that if [1,325] is colored with 2 colors, then there is a monochromatic AP of length 3. Assume indirect, and divide [1,325] into 65 blocks of length 5, say B1, . . . , B65. Each block can be colored 2 = 32 ways thus there are two blocks of the same color, Bi, Bi+d, such that 1 ≤ i < i + d ≤ 33. Within the first 3 elements of Bi two of them a, a + e are of the same color, say Red. Then all elements of the set {a, a + e, a + d, a + d + e} are Red. Note that {a, a + e, a + 2e} ⊂ Bi and {a + d, a + e + d, a + 2e + d} ⊂ Bi + d thus both a + 2e and a + 2e + d must be Blue. Now, if a + 2e + 2d is Blue, then the AP {a+2e, a+2e+d, a+2e+2d} is Blue, while if it is Red then the AP {a, a+e+d, a+2e+2d} is Red. This contradicts our indirect assumption.","PeriodicalId":179047,"journal":{"name":"The Discrete Mathematical Charms of Paul Erdős","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127418641","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Extremal Set Theory","authors":"","doi":"10.1017/9781108912181.008","DOIUrl":"https://doi.org/10.1017/9781108912181.008","url":null,"abstract":"","PeriodicalId":179047,"journal":{"name":"The Discrete Mathematical Charms of Paul Erdős","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116104377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Delta-Systems","authors":"","doi":"10.1017/9781108912181.007","DOIUrl":"https://doi.org/10.1017/9781108912181.007","url":null,"abstract":"","PeriodicalId":179047,"journal":{"name":"The Discrete Mathematical Charms of Paul Erdős","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132367932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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