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 ploglogn → ∞ and ε > 0 then, almost surely,
We also prove that there are constants c 1, c 2 such that, if plogn → 0 and p ≥ logn ∕ n, then
We give some bounds for various other ranges of p(n), but several questions are left open.
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Bollobás, B., Brightwell, G. (2013). The Dimension of Random Graph Orders. In: Graham, R., Nešetřil, J., Butler, S. (eds) The Mathematics of Paul Erdős II. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7254-4_5
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DOI: https://doi.org/10.1007/978-1-4614-7254-4_5
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