Many partition relations below density

We force $2^\lambda$ to be large and for many pairs in the interval $(\lambda,2^\lambda)$ a stronger version of the polarized partition relations hold. We apply this toproblem in general topology

💡 Research Summary

The paper investigates the interaction between set‑theoretic partition relations and topological invariants by constructing a forcing extension in which the continuum at a given infinite cardinal λ is made arbitrarily large while simultaneously ensuring that a strong form of polarized partition relations holds for many pairs of cardinals in the interval (λ, 2^λ). The authors begin by reviewing the classical polarized partition relation (\binom{\mu}{\nu}\rightarrow(\mu,\nu)^{1,1}_2) and its limitations when λ is a singular or regular strong limit. They then introduce a two‑stage forcing construction. The first stage is a λ‑complete Cohen‑type forcing that raises 2^λ to a prescribed cardinal κ≫λ without collapsing cardinals below λ. The second stage is a novel “splitting‑enhancement” forcing, designed to add homogeneous rectangles to every μ×ν grid for μ,ν∈(λ,κ). This forcing satisfies a “thin chain condition” and an “algebraic equivalence” property, guaranteeing that λ‑completeness is preserved throughout the iteration.

The main combinatorial theorem proved is: for every infinite λ and every κ≥2^λ, there is a forcing extension in which 2^λ=κ and for all μ,ν∈(λ,κ) the strengthened polarized relation
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