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Throughout this abstract let U be a fixed p-point ultrafilter and let I bethe dual ideal. Grigorieff forcing is P(U)={p:omega to 2|dom(p) is an elementof I} ordered by reverse inclusion. It is well known that Grigorieff forcing isproper. The main result of this paper is the following: THEOREM: Gregorieff forcing does not satisfy Axiom A. To prove this we use thefollowing game, denoted G(U), for two players playing alternatively: Player I plays a partition of omega, {J_n| n<omega}, such that for alln<omega, J_n is an element of I;  At the nth turn Player II plays a finite subset F_n of J_n.Player II wins iff the union of the F_n is an element of U.  The following two Lemmas prove the Theorem: LEMMA 1: If P(U) satisfies axiom A, then player II has a winning strategyin the game G(U). LEMMA 2:The game G(U) is undetermined.

Combinatorics on ideals and axiom A