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 < 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 c1p−1≤dimPn, p≤c2p−1. We give some bounds for various other ranges of p(n), but several questions are left open.
The Mathematics of Paul Erdos II, Second Edition
Bollobás, B., & Brightwell, G. (2013). The dimension of random graph orders. The Mathematics of Paul Erdos II, Second Edition, 47-68. https://doi.org/10.1007/978-1-4614-7254-4_5