Planar Ramsey Numbers

The planar Ramsey number PR(k, l) (k, l ≥ 2) is the smallest integer n such that any planar graph on n vertices contains either a complete graph on k vertices or an independent set of size l. We find exact values of PR(k, l) for all k and l. Included is a proof of a 1976 conjecture due to Albertson, Bollobás, and Tucker that every triangle-free planar graph on n vertices contains an independent set of size ⌊n/3⌋ + 1.