Open Coloring Axioms and Combinatorics of the Reals

Ramsey's theorem, that every (2-)coloring of the naturals has an infinite homogeneous set, is a fundamental result in combinatorics. Given its wide applicability, a natural question is whether there are generalizations thereof to the uncountable. However, the naive generalization is false (at least in ZFC), and therefore set theorists have investigated generalizations that place topological restraints on the colorings. These are known as Open Coloring Axioms.

In this talk, we will introduce the two main Open Coloring Axioms and prove some results about their impact on the size of the bounding number. If time permits, we will prove that the conjunction of the two (which is consistent!) decides the value of the continuum.