In mathematical set theory, 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 follows from the proper forcing axiom.
Statement
Suppose that X is a subset of the reals, and each pair of elements of X is colored either black or white, with set of white pairs being open. The open coloring axiom states that either X has an uncountable subset such that any pair from this subset is white, or X can be partitioned into a countable number of subsets such that any pair from the same subset is black.
References
Open coloring axiom Wikipedia(Text) CC BY-SA