Some remarks about embeddings of l ₁ ^k in finite-dimensional spaces

The span X _n of functions x _i(t)=±1, i=1, ..., n, on a set T in the supremum norm is considered. It is proved, for example, that X _n contains an isometric copy of l ₁ ^k for k≧cM _n ² /n log n where M _n is the Rademacher average of {x _i} ₁ ⁿ . This generalizes a result of Pisier for characters. The proof uses a new combinatorial tool.