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The Dimension of Random Graph Orders

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Part of the Algorithms and Combinatorics book series (AC,volume 14)

Summary

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.

Keywords

  • 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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© 1997 Springer-Verlag Berlin Heidelberg

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Bollobás, B., Brightwell, G. (1997). The Dimension of Random Graph Orders. In: Graham, R.L., Nešetřil, J. (eds) The Mathematics of Paul Erdös II. Algorithms and Combinatorics, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60406-5_6

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  • DOI: https://doi.org/10.1007/978-3-642-60406-5_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-64393-4

  • Online ISBN: 978-3-642-60406-5

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