We prove that if X is a subspace of Lp (2 < p < ), then either X embeds isomorphically into p 2 or X contains a subspace Y, which is isomorphic to p(.2). We also give an intrinsic characterization of when X embeds into p 2 in terms of weakly null trees in X or, equivalently, in terms of the "infinite asymptotic game" played in X. This solves problems concerning small subspaces of Lp originating in the 1970's. The techniques used were developed over several decades, the most recent being that of weakly null trees developed in the 2000's.
