This is an announcement for the paper "Small subspaces of L_p" by R.Haydon, E.Odell, Th.Schlumprecht. Abstract: We prove that if $X$ is a subspace of $L_p$ $(2<p<\infty)$ then either $X$ embeds isomorphically into $\ell_p \oplus \ell_2$ or $X$ contains a subspace $Y$, which is isomorphic to $\ell_p(\ell_2)$. We also give an intrinsic characterization of when $X$ embeds into $\ell_p \oplus \ell_2$ 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. Archive classification: math.FA Mathematics Subject Classification: 46E30 The source file(s), smallsubspaces.tex: 99982 bytes, is(are) stored in gzipped form as 0711.3919.gz with size 31kb. The corresponding postcript file has gzipped size 185kb. Submitted from: schlump@math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0711.3919 or http://arXiv.org/abs/0711.3919 or by email in unzipped form by transmitting an empty message with subject line uget 0711.3919 or in gzipped form by using subject line get 0711.3919 to: math@arXiv.org.