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 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.