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:
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.
A weaker version, OCAP, replaces the uncountability condition in the first case with being a compactperfect set in X. Both OCA and OCAP can be stated equivalently for arbitrary separable spaces.
OCA implies that the smallest unbounded set of Baire space has cardinality . Moreover, assuming OCA, Baire space contains few "gaps" between sets of sequences — more specifically, that the only possible gaps are (ω1,ω1)-gaps and (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, Zbl0585.03019
Carotenuto, Gemma (2013), An introduction to OCA(PDF), notes on lectures by Matteo Viale