Open coloring axiom

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

• Abraham, Uri; Rubin, Matatyahu; Shelah, Saharon (1985), "On the consistency of some partition theorems for continuous colorings, and the structure of ℵ₁-dense real order types", Ann. Pure Appl. Logic, 29: 123–206, doi:10.1016/0168-0072(84)90024-1, Zbl 0585.03019 
• Carotenuto, Gemma (2013), An introduction to OCA (PDF), notes on lectures by Matteo Viale 
• Kunen, Kenneth (2011), Set theory, Studies in Logic, 34, London: College Publications, ISBN 978-1-84890-050-9, Zbl 1262.03001 
• Moore, Justin Tatch (2011), "Logic and foundations the proper forcing axiom", in Bhatia, Rajendra, Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol. II: Invited lectures (PDF), Hackensack, NJ: World Scientific, pp. 3–29, ISBN 978-981-4324-30-4, Zbl 1258.03075 
• Todorčević, Stevo (1989), Partition problems in topology, Contemporary Mathematics, 84, Providence, RI: American Mathematical Society, ISBN 0-8218-5091-1, MR 0980949, Zbl 0659.54001