Let be a property of graphs on a fixed n-element vertex set V. The complexity c() is the minimal number of edges whose existence in a previously unknown graph H has to be tested such that it becomes possible to decide whether H has property or not. We investigate properties (in the broader sense of families of sets) whose complexity is much less than the maximum (which is for graph properties). It is well known that such properties must be representable as a disjoint union of long intervals (in the boolean lattice of graphs on V). We show that, if the number of intervals is not too large, the converse is true as well. We also show that there are graph properties whose complexities differ by at most 4n from any given number between 2n − 4 and . Finally, we give estimates on the complexity of the scorpion graph property.