If λ ≥ 2^κ, then for every club subset D, the ideal Iκ,λ restricted to D is not (λ,2)-distributive and fails the partition relation J+ J → (J+)^2_ω.
CARR, Pκλ Partition relations , Fundamenta Mathematicae 128 (1987), 181-195
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Piece selection and cardinal arithmetic
If λ ≥ 2^κ, then for every club subset D, the ideal Iκ,λ restricted to D is not (λ,2)-distributive and fails the partition relation J+ J → (J+)^2_ω.