We show that for every integer n\geq 1 there exists a graph G_n with n ^{1 + o(1)} edges such that every n-vertex planar graph is isomorphic to a subgraph of G_n. 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. This significantly reduces the number of edges in a recent construction of the authors with Dujmovi\'c, Gavoille, and Micek.}
We show that for every integer n\geq 1 there exists a graph G_n with (1+o(1))n vertices and n ^{1 + o(1)} edges such that every n-vertex planar graph is isomorphic to a subgraph of G_n. The best previous bound on the number of edges was O(n^{3/2}), proved by Babai, Chung, Erdos, 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. This significantly reduces the number of edges in a recent construction of the authors with Dujmovi\'c, Gavoille, and Micek.}
R
n
(1+o(1))n vertices and n
A
Chung,
D
Chung,
A
We then show that for every integer n\geq 1 there is a graph U_n with n^{1 + o(1)