Embedding of l(Formula presented.) in finite dimensional Banach spaces

Let x₁, x₂, ..., x_n be n unit vectors in a normed space X and define M_n=Ave{‖Σ(Formula presented.)ε₁x_i‖:ε₁=±1}. We prove that there exists a set A⊂{1, ..., n} of cardinality(Formula presented.) such that {x_i}_i∈A is 16 M_n-isomorphic to the natural basis of l(Formula presented.). This result implies a significant improvement of the known results concerning embedding of l(Formula presented.) in finite dimensional Banach spaces. We also prove that for every ∈>0 there exists a constant C(∈) such that every normed space X_n of dimension n either contains a (1+∈)-isomorphic copy of l(Formula presented.) for some m satisfying ln ln m≧1/2 ln ln n or contains a (1+∈)-isomorphic copy of l(Formula presented.) for some k satisfying ln ln k>1/2 ln ln n−C(∈). These results follow from some combinatorial properties of vectors with ±1 entries.