Strong Compactness and a Partition Property

We show that if Part(κ, λ) holds for every λ ≥ κ, then κ is strongly compact. Let κ be a regular infinite cardinal, and let λ ≥ κ be a cardinal. Pκ(λ) denotes the set of all subsets of λ of size less than κ. Part(κ, λ) means that for every F : Pκ(λ) × Pκ(λ) → 2, there is a cofinal subset A of (Pκ(λ),⊆) such that F is constant on the set {(a, b) ∈ A × A : a ⊂ b}. This definition is due to Jech [4]. Jech and Shelah [5] established that Part(κ, κ) holds for κ = ω. We proved in [10] that if κ is almost λ-ineffable, then Part(κ, λ) holds. It is also known ([5], [11], [12], [8]) that if κ is mildly λ-ineffable and cov(Mκ,λ<κ) > λ, then Part(κ, λ) holds. Let μ ≥ κ be a cardinal. We will show that if Part(κ, 22μ <κ ) holds, then κ is μ-compact. First we recall a few definitions. Given a cardinal ν ≥ κ, Iκ,ν denotes the set of all A ⊆ Pκ(ν) such that {a ∈ A : b ⊆ a} = ∅ for some b ∈ Pk(ν). By an ideal on Pκ(ν) we mean a subset K of P (Pκ(ν)) such that (i) Iκ,ν ⊆ K, (ii) Pκ(ν) ∈ K, (iii) P (A) ⊆ K for every A ∈ K, and (iv) ⋃ T ∈ K for every subset T of K of size less than κ. We let K = P (Pκ(ν))\K. For A ∈ K, we let K|A = {B ⊆ Pκ(ν) : B ∩ A ∈ K}. For a cardinal τ ≥ 2, K is τ -saturated if there is no size τ subset T of K with the property that A ∩ B ∈ K for any two distinct members A,B of T . K is nowhere τ -saturated if for every A ∈ K, K|A is not τ -saturated. K is prime if it is 2-saturated. We say that κ is ν-compact if there exists a prime ideal on Pκ(ν). The following is due to Jech [3]. Lemma 1. If Part(κ, λ) holds, then κ is weakly compact. The following is due to Levy and Silver (see [6], Proposition 16.4(b)). Lemma 2. Let J be an ideal on Pκ(μ). If J is κ-saturated and κ is weakly compact, then J |A is prime for some A ∈ J. Let Jκ,μ denote the collection of all nowhere κ-saturated ideals on Pκ(μ). Note that an ideal J on Pκ(μ) belongs to Jκ,μ if and only if for every A ∈ J, there is a partition 〈Dξ : ξ < κ〉 of A such that Dξ ∈ J for every ξ < κ. The following is essentially due to Taylor (see [13], Theorem 2.2). Received by the editors June 22, 2004 and, in revised form, February 11, 2005. 2000 Mathematics Subject Classification. Primary 03E02, 03E55.