The dimension of random graph orders

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 \u3c 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 \u3e 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 c1p−1≤dimPn, p≤c2p−1. We give some bounds for various other ranges of p(n), but several questions are left open