  We study some properties of graphs (or, rather, graph sequences) defined by
demanding that the number of subgraphs of a given type, with vertices in
subsets of given sizes, approximatively equals the number expected in a random
graph. It has been shown by several authors that several such conditions are
quasi-random, but that there are exceptions. In order to understand this
better, we investigate some new properties of this type. We show that these
properties too are quasi-random, at least in some cases; however, there are
also cases that are left as open problems, and we discuss why the proofs fail
in these cases.
  The proofs are based on the theory of graph limits; and on the method and
results developed by Janson (2011), this translates the combinatorial problem
to an analytic problem, which then is translated to an algebraic problem.
