During a recent workshop, the question came up whether there are some constructions for graphs that are induced $C_5$-free, but they contain "everything else," so we don't want to forbid $C_5$'s, other induced cycles and so on. What are some non-trivial constructions for such graphs? I suppose the question also makes sense for other graphs.
Just to clarify, by construction I mean something very explicit, like the vertices correspond to length $n$ zero-one sequences and we connect two vertices if their inner product is a prime number.