The open coloring axiom (abbreviated OCA) is an axiom about coloring edges of a graph whose vertices are a subset of the real numbers: two different versions were introduced by Abraham, Rubin & Shelah (1985) and by Todorčević (1989).
The open coloring axiom states that either: A weaker version, OCAP, replaces the uncountability condition in the first case with being a compact perfect set in X.
OCAP can be proved in ZFC for analytic subsets of a Polish space, and from the axiom of determinacy.
The full OCA is consistent with (but independent of) ZFC, and follows from the proper forcing axiom.
OCA implies that the smallest unbounded set of Baire space has cardinality