Embeddingl p k in subspaces ofL p forp>2

The aim of the present paper is to estimate in a precise manner the integerk=k(p,m,n,∈) so that an arbitrarym-dimensional subspaceX of the spacel _p ⁿ ;p>2, contains an (1+∈)-isomorph ofl _p ^k . The main argument of the proof consists of a probabilistic selection which uses a lemma of Slepian. The same method also shows that any system of normalized functions inL _p ;p≥2, which is equivalent to the unit vector basis ofl _p ⁿ , contains, for any∈>0, a subsystem of sizeh proportional ton, which is (1+∈)-equivalent to the unit vector basis ofl _p ^h . The authors were supported by Grant No. 87-0079 from BSF.