Small Subspaces of L_p

We prove that if $X$ is a subspace of $L_p$ $(2<p<\mathrm{\infty })$, then either $X$ embeds isomorphically into ${\mathrm{\ell }}_p\oplus {\mathrm{\ell }}_2$ or $X$ contains a subspace $Y,$ which is isomorphic to ${\mathrm{\ell }}_p({\mathrm{\ell }}_2)$. We also give an intrinsic characterization of when $X$ embeds into ${\mathrm{\ell }}_p\oplus {\mathrm{\ell }}_2$ 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 $L_p$ 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.