Abstract
We investigate the partition property of \({\mathcal{P}_{\kappa}\lambda}\). Main results of this paper are as follows: (1) If λ is the least cardinal greater than κ such that \({\mathcal{P}_{\kappa}\lambda}\) carries a (λ κ, 2)-distributive normal ideal without the partition property, then λ is \({\Pi^1_n}\)-indescribable for all n < ω but not \({\Pi^2_1}\) -indescribable. (2) If cf(λ) ≥ κ, then every ineffable subset of \({\mathcal{P}_{\kappa}\lambda}\) has the partition property. (3) If cf(λ) ≥ κ, then the completely ineffable ideal over \({\mathcal{P}_{\kappa}\lambda}\) has the partition property.
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Abe, Y., Usuba, T. Notes on the partition property of \({\mathcal{P}_\kappa\lambda}\) . Arch. Math. Logic 51, 575–589 (2012). https://doi.org/10.1007/s00153-012-0283-x
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DOI: https://doi.org/10.1007/s00153-012-0283-x