  We consider rooted subgraphs in random graphs, i.e., extension counts such as
(i) the number of triangles containing a given vertex or (ii) the number of
paths of length three connecting two given vertices. In 1989, Spencer gave
sufficient conditions for the event that, with high probability, these
extension counts are asymptotically equal for all choices of the root vertices.
For the important strictly balanced case, Spencer also raised the fundamental
question as to whether these conditions are necessary. We answer this question
by a careful second moment argument, and discuss some intriguing problems that
remain open.
