  FI-graphs were introduced by the second author and White to capture the idea
of a family of nested graphs, each member of which is acted on by a
progressively larger symmetric group. That work was built on the newly minted
foundations of representation stability theory and FI-modules. Examples of such
families include the complete graphs and the Kneser and Johnson graphs, among
many others. While it was shown in the originating work how various counting
invariants in these families behave very regularly, not much has thus far been
proven about the behaviors of the typical extremal graph theoretic invariants
such as their independence and clique numbers. In this paper we provide a
conjecture on the growth of the independence and clique numbers in these
families, and prove this conjecture in one case. We also provide computer code
that generates experimental evidence in many other cases. All of this work
falls into a growing trend in representation stability theory that displays the
regular behaviors of a number of extremal invariants that arise when one looks
at FI-algebras and modules.
