Before revision:  The best previous bound on the number of edges was O(n^{3/2}), proved by Babai,  Erdos, Chung, Graham, and Spencer in 1982.  vertices and edges that contains induced copies of every n-vertex planar graph.
After revision:  The best previous bound on the number of edges was O(n^{3/2}), proved by Babai, Erdos, Chung, Graham, and Spencer in 1982. We then show that for every integer n\geq 1 there is a graph U_n with n^{1 + o(1) vertices and edges that contains induced copies of every n-vertex planar graph.