Planar Ramsey Numbers

Abstract 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, Bollobas, and Tucker that every triangle-free planar graph on n vertices contains an independent set of size ⌊ n /3⌋ + 1.