The dimension of sparse and co-sparse random graph orders

Abstract

A random graph order is a partial order obtained from a random graph on $[n]$ by taking the transitive closure of the adjacency relation. The dimension of the random graph orders from random bipartite graphs $B(n,n,p)$ and from $G(n,p)$ were previously studied when $p=Ω(\log n/n)$ and when $p$ is not too close to 1. There is a conjectured phase transition in the sparse range at $p=1/n$. In this paper, we investigate this conjectured phase transition and estimate the dimension of the partial orders arising from $B(n,n,p)$ and $G(n,p)$ when $p=O(1/n)$. For the random bipartite order, we additionally estimate its dimension in the co-sparse regime, thereby closing all previously open ranges of $p$. Finally, we establish a general upper bound on the dimension of partial orders based on their decompositions into suborders, a result that is of independent interest.