Counting extensions in random graphs

We consider rooted subgraphs in random graphs, i.e., extension counts such as (a) the number of triangles containing a given ‘root’ vertex, or (b) the number of paths of length three connecting two given ‘root’ vertices.
In 1989 Spencer gave sufficient conditions for the event that, whp, all roots of the binomial random graph G(n,p) have the same asymptotic number of extensions, i.e., $(1 \pm \varepsilon)$ times their expected number. For the important strictly balanced case, Spencer also raised the fundamental question whether these conditions are necessary. We answer this question by a careful second moment argument, and discuss some open problems and cautionary examples for the general case.
Based on joint work with Matas Sileikis.