Polylog Dimensional Subspaces of l(infinity)(N)

We show that a subspace of l(infinity)(N) of dimension n > (logN log log N)(2) contains 2-isomorphic copies of l(infinity)(k) where k tends to infinity with n/(logN log logN)(2). More precisely, for every eta > 0, we show that any subspace of l(infinity)(N) of dimension n contains a subspace of dimension m = c(eta)root n/(logN log logN) of distance at most 1 + eta from l(infinity)(m).