Creating Small Subgraphs in Achlioptas Processes With Growing Parameter

We study the problem of creating a copy of some fixed graph H in the Achlioptas process on n vertices with parameter r, where r = r(n) is a growing function of n. We prove general upper and lower bounds on the threshold of this problem, and derive exact threshold functions for the case where H is a tree, a cycle, or the complete graph on four vertices.