Abstract. We show a general scheme of Ramsey-type results for partitions of countable sets of finite functions, where "one piece is big" is interpreted in the language originating in creature forcing. The heart of our proofs follows Glazer's proof of the Hindman Theorem, so we prove the existence of idempotent ultrafilters with respect to suitable operation. Then we deduce partition theorems related to creature forcings.
IntroductionA typical partition theorem asserts that if a set with some structure is divided into some number of "nice" pieces, then one of the pieces is large from the point of the structure under considerations. Sometimes, the underlying structure is complicated and it is not immediately visible that the arguments in hands involve a partition theorem. Such is the case with many forcing arguments. For instance, the proofs of propernes of some forcing notions built according to the scheme of norms on possibilities have in their hearts partition theorems stating that at some situations a homogeneous tree and/or a sequence of creatures determining a condition can be found (see, e.g., Ros lanowski and Shelah [7,8], Ros lanowski, Shelah and Spinas [9], Kellner and Shelah [6,5]). A more explicit connection of partition theorems with forcing arguments is given in Shelah and Zapletal [10].The present paper is a contribution to the Ramsey theory in the context of finitary creature forcing. We are motivated by earlier papers and notions concerning norms on possibilities, but we do not look at possible forcing consequences. The common form of our results here is as follows. If a certain family of partial finite functions is divided into finitely many pieces, then one of the pieces contains all partial functions determined by an object ("a pure candidate") that can be interpreted as a forcing condition if we look at the setting from the point of view of the creature forcing. Sets of partial functions determined by a pure candidate might be considered as "large" sets.Our main proofs are following the celebrated Glazer's proof of the Hindman Theorem, which reduced the problem to the existence of a relevant ultrafilter on ω in ZFC. Those arguments were presented by Comfort in [2, Theorem 10.3, p.451] with [2, Lemma 10.1, p.449] as a crucial step (stated here in 2.7). The arguments of the second section of our paper really resemble Glazer's proof. In that section we deal with the easier case of omittory-like creatures (loose FFCC pairs of 1.2(2))