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.
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Communicated by W. T. Trotter
Supported in part by NSF under Grant No. MCS83-01867 and by a Sloan Research Fellowship.
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Füredi, Z., Kahn, J. On the dimensions of ordered sets of bounded degree. Order 3, 15–20 (1986). https://doi.org/10.1007/BF00403406
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DOI: https://doi.org/10.1007/BF00403406