TITLE: Fáry's theorem for infinite graphs
QUESTION [5 upvotes]: Fáry's theorem is a (fairly famous) statement which asserts that every finite simple planar graph can be drawn in a way such that every edge is represented by a straight line segment.

Does this theorem extend to countably infinite graphs?

A standard proof of this theorem proceeds by induction on the number of vertices, and hence is unapplicable here. I suspect the answer is no, but I can't think of a counterexample.
Thanks in advance.

REPLY [2 votes]: It looks like this problem was solved. I found a paper proving that every infinite planar graph has a straight line representation.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.102.1891&rep=rep1&type=pdf