Before revision:  In particular, in the vertex-arrival streams, we get a randomized O( \frac{\sqrt{n}{\eps^2}} }{\epsilon^2}} \log n) space algorithm for approximating the matching size within (3.5+   ) factor in a planar graph on n vertices.
After revision:  In particular, in the vertex-arrival streams, we get a randomized O( \frac{\sqrt{n}{\eps^2}} \frac{\sqrt{n}{\epsilon^2}} \log n) space algorithm for approximating the matching size within (3.5+  ) factor in a planar graph on n vertices.