2011-03-01, English, Article, Report edition: An infinite combinatorial statement with a poset parameter.(Report) Gillibert, Pierre; Wehrung, Friedrich

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Physical Description
  • Report
  • Magazine/​Journal article
Published
  • Springer, 2011 March
Language
  • English

Edition details

Title
  • An infinite combinatorial statement with a poset parameter.(Report)
Also Titled
  • An infinite combinatorial statement with a poset parameter.
Appears In
  • Combinatorica, v.31, no.2, 2011 March, p.183(18) (ISSN: 0209-9683)
Author
  • Gillibert, Pierre
  • Wehrung, Friedrich
Published
  • Springer, 2011 March
Physical Description
  • Report
  • Magazine/​Journal article
Subjects
Target Audience
  • Academic
Summary
  • Byline: Pierre Gillibert (1), Friedrich Wehrung (1) Keywords: 03E05; 06A07; 05A05; 05D10; 06A05 Abstract: We introduce an extension, indexed by a partially ordered set P and cardinal numbers Io,[lambda], denoted by (Io,<[lambda])aP, of the classical relation (Io,n,[lambda])[right arrow][rho] in infinite combinatorics. By definition, (Io,n,[lambda])[right arrow][rho] holds if every map F: [Io].sup.n[right arrow][Io]<[lambda] has a [rho]-element free set. For example, Kuratowski's Free Set Theorem states that (Io,n,[lambda])[right arrow]n+1 holds iff Io aY=​ [lambda] .sup.+n, where [lambda] .sup.+n denotes the n-th cardinal successor of an infinite cardinal [lambda]. By using the (Io,<[lambda])aP framework, we present a self-contained proof of the first author's result that ([lambda] .sup.+n,n,[lambda])[right arrow]n+2, for each infinite cardinal [lambda] and each positive integer n, which solves a problem stated in the 1985 monograph of ErdAs, Hajnal, Mate, and Rado. Furthermore, by using an order-dimension estimate established in 1971 by Hajnal and Spencer, we prove the relation ([lambda] .sup. + (n - 1) ,r,[lambda] ) 2.sup. \tfrac1 2(1 - 2.sup. - r ).sup. - n/​r , for every infinite cardinal [lambda] and all positive integers n and r with 2a$?r<n. For example, (au.sub.210,4,au.sub.0)[right arrow]32,768. Other order-dimension estimates yield relations such as (au.sub.109,4,au.sub.0) [right arrow] 257 (using an estimate by Furedi and Kahn) and (au.sub.7,4,au.sub.0)[right arrow]10 (using an exact estimate by Dushnik). Author Affiliation: (1) LMNO, CNRS UMR 6139 Departement de Mathematiques, BP 5186, Universite de Caen, Campus 2, 14032, Caen cedex, France Article History: Registration Date: 10/​08/​2011 Received Date: 23/​02/​2009 Online Date: 11/​08/​2011 Article note: This work was partially supported by the institutional grant MSM 0021620839.
Terms of Use
  • COPYRIGHT 2011 Springer
Language
  • English
Database
  • Gale Academic OneFile
Identifier
  • 264000708

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