The Dimension of Random Graph Orders

  • Béla Bollobás
  • Graham Brightwell
Part of the Algorithms and Combinatorics book series (AC, volume 14)


The random graph order P n, p is obtained from a random graph G n, p on [n] by treating an edge between vertices i and j, with i ≺ j in [n], as a relation i < j, and taking the transitive closure. This paper forms part of a project to investigate the structure of the random graph order P n, p throughout the range of p = p(n). We give bounds on the dimension of P n, p for various ranges. We prove that, if p log log n → ∞ and ε > 0 then, almost surely,
$$\left( {1 - \in } \right)\sqrt {\frac{{\log n}} {{\log (1/q)}}} \leqslant \dim P_{n,p} \leqslant (1 + \in )\sqrt {\frac{{4\log n}} {{3\log (1/q)}}} .$$
We also prove that there are constants c 1, c 2 such that, if p log n → 0 and p ≥ log n/n, then
$$c_1 p^{ - 1} \leqslant \dim P_{n,p} \leqslant c_2 p^{ - 1} .$$

We give some bounds for various other ranges of p(n), but several questions are left open.


Random Graph Linear Extension Transitive Closure Isoperimetric Inequality Dimension Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Béla Bollobás
    • 1
  • Graham Brightwell
    • 2
  1. 1.Department of Pure Mathematics and Statistics of MathematicsUniversity of CambridgeCambridgeUK
  2. 2.Department of Pure MathematicsLondon School of Economics and Political ScienceLondonUK

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