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  • 06A10  (3)
  • Springer  (3)
  • Blackwell Publishing Ltd
  • Springer Nature
  • 1
    Electronic Resource
    Electronic Resource
    On the dimensions of ordered sets of bounded degree (1986)
    Springer
    Order 3 (1986), S. 15-20 
    Details
    ISSN: 1572-9273
    Keywords: 06A10 ; Partially ordered set
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let P be a partially ordered set. Define k = k (P) = max p∈ |{x ∈ P : p 〈 x or p = x}|, i.e., every element is comparable with at most k others. Here it is proven that there exists a constant c (c 〈 50) such that dim P 〈 ck(log k)2. This improves an earlier result of Rödl and Trotter (dim P ≤2 k 2+2). Our proof is nonconstructive, depending in part on Lovász' local lemma.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00403406
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  • 2
    Electronic Resource
    Electronic Resource
    Balancing poset extensions (1984)
    Springer
    Order 1 (1984), S. 113-126 
    Details
    ISSN: 1572-9273
    Keywords: 06A10 ; 68E05 ; Sorting ; comparison ; information theoretic bound ; linear extension
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We show that any finite partially ordered setP (not a total order) contains a pair of elementsx andy such that the proportion of linear extensions ofP in whichx lies belowy is between 3/11 and 8/11. A consequence is that the information theoretic lower bound for sorting under partial information is tight up to a multiplicative constant. Precisely: ifX is a totally ordered set about which we are given some partial information, and ife(X) is the number of total orderings ofX compatible with this information, then it is possible to sortX using no more thanC log2 e (X) comparisons whereC is approximately 2.17.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00565647
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  • 3
    Electronic Resource
    Electronic Resource
    Dimension versus size (1988)
    Springer
    Order 5 (1988), S. 17-20 
    Details
    ISSN: 1572-9273
    Keywords: 06A10 ; 06A23 ; Lattice ; order dimension ; least size
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We investigate the behavior of f(d), the least size of a lattice of order dimension d. In particular we show that the lattice of a projective plane of order n has dimension at least n/ln(n), so that f(d)=O(d) 2 log2 d. We conjecture f(d)=θ(d 2 ), and prove something close to this for height-3 lattices, but in general we do not even know whether f(d)/d→∞.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00143893
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