  The 4 Color Theorem (4CT) implies that every $n$-vertex planar graph has an
independent set of size at least $\frac{n}4$; this is best possible, as shown
by the disjoint union of many copies of $K_4$. In 1968, Erd\H{o}s asked whether
this bound on independence number could be proved more easily than the full
4CT. In 1976 Albertson showed (independently of the 4CT) that every $n$-vertex
planar graph has an independent set of size at least $\frac{2n}9$. Until now,
this remained the best bound independent of the 4CT. Our main result improves
this bound to $\frac{3n}{13}$.
