I'm supposed to show that the number of intersection graphs of $n$ convex sets (or $n$ simple curves) in the plane is at least $2^{\Omega(n^2)}$, but I don't really know how to do this.
I know that $2^{\Omega(n^2)}$ is a lower bound for the number of simple arrangements of $n$ pseudolines, and I guess that there could be a connection between intersection graphs and these arrangements (maybe these graphs can somehow be "extended" to be arrangements?), but I don't have any concrete ideas. Other than that, I don't see another way to approach this.
I was told that this can be proved for intersection graphs of simple curves as well, but that does not help me come up with any new ideas.
So I'm just looking for any advice/hint on how to prove this.
Thanks for any help.