The Dimension of Random Graph Orders

Béla Bollobás () and Graham Brightwell
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Béla Bollobás: University of Cambridge, Department of Pure Mathematics and Mathematical Statistics
Graham Brightwell: London School of Economics and Political Science, Department of Mathematics

A chapter in The Mathematics of Paul Erdős II, 2013, pp 47-68 from Springer

Abstract: 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 0 then, almost surely, $$\displaystyle{(1+\epsilon )\sqrt{ \frac{\log n} {\log (1/q)}} \leq \dim P_{n,p} \leq (1+\epsilon )\sqrt{ \frac{4\log n} {3\log (1/q)}}.}$$ We also prove that there are constants c 1, c 2 such that, if plogn → 0 and p ≥ logn ∕ n, then $$\displaystyle{c_{1}{p}^{-1} \leq \dim P_{ n,p} \leq c_{2}{p}^{-1}.}$$ We give some bounds for various other ranges of p(n), but several questions are left open.

Keywords: Partial Order; Random Graph; Transitive Closure; Unicyclic Graph; Unrelated Pair (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/978-1-4614-7254-4_5

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