This is an announcement for the paper "On Dvoretzky's theorem for subspaces of $L_p$" by Grigoris Paouris and Petros Valettas. Abstract: We prove that for any $p > 2$ and every $n$-dimensional subspace $X$ of $L_p$, the Euclidean space $\ell_2^k$ can be $(1 + \varepsilon)$-embedded into $X$ with $k \geq c_p \min\{\varepsilon^2 n, (\varepsilon n)^{2/p} \}$, where $c_p > 0$ is a constant depending only on $p$. Archive classification: math.FA math.MG Mathematics Subject Classification: 46B07, 46B09 Remarks: 20 pages Submitted from: valettasp@missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1510.07289 or http://arXiv.org/abs/1510.07289