Many partition relations below density

We force 2^λ to be large, and for many pairs in the interval (λ, 2^λ) a strong version of the polarized partition relations holds. We apply this to problems in general topology. For example, consistently, every 2^λ is the successor of a singular and for every Hausdorff regular space X, hd(X) ≤ s(X)⁺³, hL(X) ≤ s(X)⁺³ and better when s(X) is regular, via a halfgraph partition relations. For the case s(X) = א₀ we get hd(X), hL(X) ≤ N₂.