Open Colorings, the Continuum and the Second Uncountable Cardinal

The purpose of this article is to analyze the cardinality of the continuum using Ramsey theoretic statements about open colorings or “open coloring axioms.” In particular I will show that the conjunction of two well known axioms, OCA[ARS] and OCA[T], implies that the size of the continuum is א2. Our focus in this paper will be the following two open coloring axioms 1 and their influence on the size of the continuum. OCA[ARS] If X is a separable metric space of size א1 and c : [X] → {1, . . . , n} is a continuous map then there is a decomposition of X into countably many pieces Xi (i ∈ N) such that c is constant on [Xi] for all i ∈ N. OCA[T] If X is a separable metric space and G ⊆ [X] is open then either G is countably chromatic (there is a decomposition of X into countably many pieces Xi such that [Xi] 2 ∩ G is empty for all i) or there is an uncountable H ⊆ X such that [H] ⊆ G. ∗The research for this paper was supported by EPSRC grant GR/M71121 during my stay at the University of East Anglia. I would also like to acknowledge the support I received from the Institut Mittag-Leffler during my visit there. The subscripts [ARS] and [T] refer to [1] and [11] where these axioms originally appeared. In the current literature OCA has come to mean OCA[T].