The dimension of sparse random graph orders

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 $p=\Omega(\log n/n)$, and there is a conjectured phase transition in the sparse range $p=O(1/n)$. We investigate the dimensions of the partial orders arising from $B(n,n,p)$ and $G(n,p)$ in this sparse range. We also prove an upper bound on the dimension of general partial orders based on decompositions of partial orders into suborders, which are of independent interest.