Does OCA imply the continuum is ℵ2?
Let OCA denote the assertion that every open graph on a separable metric space either has countable chromatic number or else has an uncountable clique. This principle was formulated in the 1980s by Todorcevic in light of the work of Abraham-Rubin-Shelah in which they analyzed a related principle bearing the same name. In 2001, I showed that the formulations of OCA of Abraham-Rubin-Shelah and of Todorcevic taken together imply that the continuum is ℵ2. The question of whether Todorcevic's formulation of OCA settles the value of the continuum remains an open problem. This talk will discuss some partial results and test questions.