TITLE: Graphs with few induced subgraphs
QUESTION [18 upvotes]: Which graphs have the property that the number of $i$-vertex induced subgraphs is at most $i$ for some $i < n/2$ (where $n$ is the number of vertices)?
To avoid cases I am not interested in, I want the graph to be connected and non-bipartite and its complement to have the same properties.
Of course every graph has this property for $i=2$, and every triangle-free graph (or complement of one) for $i=3$. What about larger values of $i$?
Edit. 
More background information on the problem can be found  here.

REPLY [4 votes]: Thanks, Doug. But, of course, I am not satisfied with this answer! I would like the graph to be vertex-transitive, and edge-transitive. I already know that its diameter is at most 4 (in my application).
I will come a bit cleaner about the application. This arises in a question of Joao Araujo about semigroup theory, which reduces to the following: Which groups are $(i,i+1)$-homogeneous but not $i$-homogeneous? (The condition $(i,i+1)$-homogeneous means that given any $i$-set $I$ and $j$-set $J$, there is an element of the group mapping $I$ to a subset of $J$.
Assume that $2i+1\le n$. Such a group is easily seen to be primitive and have at most three orbits on $2$-sets. If it is $2$-homogeneous, then it must be $2$-transitive, and then one has the possibility of using the classification of such groups.
It is easy to see that such a group has at most $i$ orbits on $i$-sets; so if it is not $2$-homogeneous, then all the symmetrised orbital graphs have the property of my original question.
By the way, five examples of groups with this property are known: the cyclic and dihedral groups of degree 5, and the affine groups $\mathrm{AGL}(1,7)$, $\mathrm{ASL}(2,3)$ and $\mathrm{AGL}(2,3)$ (with $n=7,9,9$ for the last three).