Weak Diamond and Open Colorings

The purpose of this article is to prove the relative consistency of certain statements about open colorings with 2א0 < 2א1 . In particular both OCA and the statement that every 1-1 function of size א1 is σ-monotonic are consistent with 2א0 < 2א1 . As a corollary we have that 2א0 < 2א1 does not admit a Pmax variation (in the presence of an inaccessible cardinal). The open coloring axiom, OCA, is a Ramsey theoretic statement relating to the real numbers. Since its introduction in [17] it has found a wealth of applications to problems closely related to the real line (see [5], [7], [13], [17], [19], and [20]). Frequently, applications of OCA require an application of MAא1 to complete the argument (see, e.g., [7], [20]). Farah gave some explanation of this phenomenon through the following theorem. Theorem 0.1. [6] OCA is relatively consistent with (1) t = א1. (2) There are no Q-sets. (3) There are two א1 dense sets of reals which are not order isomorphic. Since items 1-3 are all consequences of weak diamond (an equivalent of 2א0 < 2א1 — see [4]) this suggests the question of whether OCA is relatively consistent with weak diamond. In this paper I will show that this is indeed the case. I will also show the relative consistency of weak diamond with “Every partial 1-1 function f ⊆ R of size א1 is the union of countably many monotonic functions.” An application of the later result is that there is no Pmax variation for weak diamond (see [21] for a discussion of Pmax models and their variants). In particular there are two consequences of the bounded form of PFA which jointly imply 2א0 = 2א1 but which are each relatively consistent with 2א0 < 2א1 (one of them modulo an inaccessible cardinal). I would like to thank Ilijas Farah and Paul Larson for reading drafts of this paper and offering a number of useful suggestions. The research for this paper was supported by EPSRC grant GR/M71121 during my stay at the University of East Anglia.