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, Erd\H{o}s, 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.

翻译：我们显示,对于每整数美元1美元,就有一个G$n美元,上面有1美元+1美元+1美元,边上有1美元,上面有1美元+1美元,边上有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元1美元,上面有1美元,上面有1美元。1美元,上面有1美元。1美元,上面有1美元,上面有1美元,上面有1美元。1美元。1美元。1美元,上面有1美元,上面有1美元,上面有1美元1美元,上面有1美元。1美元,上面有1美元。1美元,上面有1美元,上面有1美元,上面有1美元,上面有1美元1美元,