Partition properties of subsets of P κ λ

Let κ > ω be a regular cardinal and λ > κ a cardinal. The following partition property is shown to be consistent relative to a supercompact cardinal: For any f : ⋃ n<ω [X] ⊂ → γ with X ⊂ Pκλ unbounded and 1 < γ < κ there is an unbounded Y ⊂ X with |f“[Y ]⊂| = 1 for any n < ω. Let κ be a regular cardinal > ω, λ a cardinal ≥ κ and F a filter on Pκλ. Partition properties of the form Pκλ→ (F)2 (see below for the definition) were introduced by Jech [6]. The case where F is the club filter Cκλ was particularly studied in connection with a supercompact cardinal: Menas [14] proved Pκλ → (C κλ)2 for a 2 <κ -supercompact κ via a normal ultrafilter U with Pκλ → (U)2. As noted by Kamo [9], Menas’ argument can be modified to give the partition property of Pκλ for κ just λ-supercompact. For the converse direction Di Prisco and Zwicker [4] and others refined the global result of Magidor [12]: The partition property of Pκ2λ<κ implies that κ is λ-supercompact. In [8] Johnson introduced properties of the form X → (F)2 for X ∈ F+, which means that for any f : [X]⊂ → 2 there is Y ∈ F+ with Y ⊂ X and |f“[Y ]⊂| = 1, as well as F+ → (F)2, which means X → (F)2 for any X ∈ F+. Abe [1] asked whether F κλ → (F κλ)2 would fail in ZFC, where Fκλ denotes the minimal fine filter on Pκλ. In this note we answer the question of Abe: Theorem. Let κ be a supercompact cardinal and λ a cardinal > κ. Then there is a κ+-c.c. poset forcing that κ is supercompact and F κλ → (F κλ) γ for any 1 < γ < κ. 1991 Mathematics Subject Classification: 03E05, 03E55. The author enjoyed Prof. Kanamori’s hospitality while staying at Boston University as a Japanese Overseas Research Fellow. He also wishes to thank Prof. Abe for helpful conversations, and the referee for improving the presentation.