The Dimension of Random Graph Orders

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,

$$\displaystyle{(1+\epsilon )\sqrt{ \frac{\log n} {\log (1/q)}} \leq \dim P_{n,p} \leq (1+\epsilon )\sqrt{ \frac{4\log n} {3\log (1/q)}}.}$$

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.