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  • 1
    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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    Paper (German National Licenses)
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  • 2
    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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    Paper (German National Licenses)
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