The complexity of recognizing optimal set systems for the V-C dimension
This doesn't answer the question, but it might be helpful.  Mossel and Umans have made a detailed study of the complexity of approximating VC-dimension, when the set system is succinctly presented:

On the Complexity of Approximating the VC Dimension
http://users.cms.caltech.edu/~umans/papers/MU01-final.ps