Let us say that a graph class has the jump property, if it either contains all $n$-vertex graphs for every large enough $n$, or else the fraction of $n$-vertex graphs that belong to the class approaches 0, as $n\rightarrow\infty$.
The answers to an earlier question (see here) show that every hereditary graph class has this jump property.
Question: What are some other (nontrivial) examples of graph classes with this property?
Note: It would be good to see natural examples, that is, classes that arise from independent considerations, not created just for the sake of the jump property.