Global properties of hereditary classes? A hereditary class of structures (e.g. graphs) is one that is closed under induced substructures, or equivalently, is closed under vertex removal.

Classes of graphs that exclude a minor have nice properties that do not depend on the specific excluded minor.  Martin Grohe showed that for graph classes excluding a minor there is a polynomial algorithm for isomorphism, and fixed-point logic with counting captures polynomial time for these graph classes.  (Grohe,   Fixed-Point Definability and Polynomial Time on Graphs with Excluded Minors, LICS, 2010.)  These can be thought of as "global" properties.


  Are there similar "global" properties known for hereditary classes (either graphs or more general structures)?


It would be good to see each answer focus on just one specific property.
Hereditary properties are very "robust" in the following sense.

Noga Alon and Asaf Shapira showed that for any hereditary property ${\cal P}$, if a graph $G$ needs more than $\epsilon n^2$ edges to be added or removed in order to satisfy ${\cal P}$, then there is a subgraph in $G$, of size at most $f_{\cal P}(\epsilon)$, which does not satisfy ${\cal P}$.  Here, the function $f$ only depends on the property ${\cal P}$ (and not on the size of the graph $G$, for instance).  Erdős had made such a conjecture only about the property of $k$-colorability.

Indeed, Alon and Shapira prove the following stronger fact: given ${\cal P}$, for any $\epsilon$ in $(0,1)$, there are $N(\epsilon)$, $h(\epsilon)$ and $\delta(\epsilon)$ such that if a graph $G$ has at least $N$ vertices and needs at least $\epsilon n^2$ edges added/removed in order to satisfy ${\cal P}$, then for at least $\delta$ fraction of induced subgraphs on $h$ vertices, the induced subgraph violates ${\cal P}$.  Thus, if $\epsilon$ and the property ${\cal P}$ are fixed, in order to test if an input graph satisfies ${\cal P}$ or is $\epsilon$-far from satisfying ${\cal P}$, then one only needs to query the edges of a random induced subgraph of constant size from the graph and check if it satisfies the property or not.  Such a tester would always accept graphs satisfying ${\cal P}$ and would reject graphs $\epsilon$-far from satisfying it with constant probability.  Furthermore, any property that is one-sided testable in this sense is a hereditary property!  See the paper by Alon and Shapira for details.