Large normal ideals concentrating on a fixed small cardinality

The property on the filter in Definition 1, a kind of large cardinal property, suffices for the proof in Liu Shelah LiSh484] and is proved consistent as required there (see Conclusion 6). A natural property which looks better, not only is not obtained here, but is shown to be false (in Claim 7). On earlier related theorems see Gitik Shelah GiSh310]. On such games see e.g. Je], Sh-b], Sh-f]. Received June 27, 1994