A relative of the approachability ideal, diamond and non-saturation

Let denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that * together with 2 = + implies s for every S ? + that reflects stationarily often. In this paper, for a set S ? +, a normal subideal of the weak approachability ideal is introduced, and denoted by I[S; ]. We say that the ideal is fat if it contains a stationary set. It is proved: 1. ifI[S; a] is fat, thenNS^ \ S is non-saturated; 2. if I[S; ] is fat and A = +, then Os holds; 3. * implies that I[S; ] is fat for every S ? + that reflects stationarily often; 4. it is relatively consistent with the existence of a supercompact cardinal that * fails, while I[S; ] is fat for every stationary S ? + that reflects stationarily often. The stronger principle *+ is studied as well. ?0. Introduction. 0.1. Background. Recall Jensen's diamond principle [11]: for an infinite cardinal and a stationary set S ? A+, Os asserts the existence of a collection {As \ e S} such that for all ? A+, the set { e S \ = As} is stationary. It is easy to see that 0^+ implies 2 = A+. It is then natural to ask whether the converse holds, as well. For = the answer is negative (see [3]). However, for an uncountable cardinal , a recent theorem by Shelah [22] states that 2 = + is indeed equivalent to 0a+? To state Shelah's theorem in its most general form, we need the following piece of notation. For ordinals K are defined analogously. Shelah's theorem reads as follows. Theorem (Shelah, [22]). Suppose is an uncountable cardinal and 2 = +. If S ? E^c{^ is stationary, then Os holds. Now, for a regular cardinal , the assumption 2 = + is consistent together with the negation of 0^+ (for = , see [3]; for > , see [23]), but for a singular cfU) cardinal , it is not even known whether <} + is implied by the full GCH. ^cfU) Received March 22, 2009. 2000 Mathematics Subject Classification. Primary 03E35; Secondary 03E05.