Open coloring axiom

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) and by (Todorčević 1989).

Statement

Suppose that X is a subset of the reals, and each pair of elements of X is colored either black or white, with the set of white pairs being open in the complete graph on X. The open coloring axiom states that either:

1. X has an uncountable subset such that any pair from this subset is white; or
2. X can be partitioned into a countable number of subsets such that any pair from the same subset is black.

A weaker version, OCA_P, replaces the uncountability condition in the first case with being a compact perfect set in X. Both OCA and OCA_P can be stated equivalently for arbitrary separable spaces.

Relation to other axioms

OCA_P 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 [math]\displaystyle{ \aleph_2 }[/math]. Moreover, assuming OCA, Baire space contains few "gaps" between sets of sequences — more specifically, that the only possible gaps are Hausdorff gaps and analogous (κ,ω)-gaps where κ is an initial ordinal not less than ω₂.

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 (2): 123–206, doi:10.1016/0168-0072(84)90024-1 
• Carotenuto, Gemma (2013), An introduction to OCA, notes on lectures by Matteo Viale, http://www.logicatorino.altervista.org/matteo_viale/OCA.pdf 
• Kunen, Kenneth (2011), Set theory, Studies in Logic, 34, London: College Publications, ISBN 978-1-84890-050-9 
• 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, Hackensack, NJ: World Scientific, pp. 3–29, ISBN 978-981-4324-30-4, http://www.math.cornell.edu/~justin/Ftp/ICM.pdf 
• Todorčević, Stevo (1989), Partition problems in topology, Contemporary Mathematics, 84, Providence, RI: American Mathematical Society, ISBN 0-8218-5091-1, https://archive.org/details/partitionproblem0000todo