Finite metric spaces needing high dimension for lipschitz embeddings in banach spaces

We construct a sequence of metric spaces (M _n) with cardM _n=3n satisfying that for everyc<2, there exists a real numbera(c)>0 such that, if the Lipschitz distance fromM _n to a subset of a Banach spaceE is less thanc, then dim(E) ≥a(c)n. We also prove several results about embeddings of metric spaces whose non-zero distance values are in the interval [1,2].