Narrow Results

The purpose of this article is to prove the relative consistency of certain statements about open colorings with 2^ℵ₀ < 2^ℵ₁. In particular both OCA and the statement that every 1–1 function of size ℵ₁ is σ-monotonic are consistent with 2^ℵ₀ < 2^ℵ₁. As a corollary we have that 2^ℵ₀ < 2^ℵ₁ does not admit a ℙ_max variation (in the presence of an inaccessible cardinal).