Polylog dimensional subspaces of ` N ∞ ∗

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