On the order dimension of 1-sets versus k-sets

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Abstract

A family F of s-subsets of [t] is a (θ, s, t)-family iff the intersection of any two distinct elements of F has cardinality less than θ. Let f(θ, s, t) be the greatest integer n such that there exists an (θ, s, t)-family of cardinality n. Let dim(1, k; n) denote the dimension of Bn(1, k), the suborder of the Boolean lattice on [n] consisting of 1-subsets and k-subsets of [n]. We use upper and lower bounds on f(θ, s, t) to derive new lower and upper bounds on dim(1, k; n). In particular we answer a question of Trotter by showing that dim(1, log n; n) = Ω(log3 n/log log n). The estimation of dim(1, log n; n) plays a critical role in the determination of the maximum dimension of an ordered set with fixed maximum degree. Previously it was only known that (log2 n/4 < dim(1, log n; n) <log3 n.

Original languageEnglish (US)
Pages (from-to)219-228
Number of pages10
JournalJournal of Combinatorial Theory. Series A
Volume73
Issue number2
DOIs
StatePublished - Jan 1 1996

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ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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