Notes on the partition property of {\ mathcal {P} _\ kappa\ lambda}

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