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Ramseyan ultrafilters

We investigate families of partitions of omega which are related to special coideals, so-called happy families, and give a dual form of Ramsey ultrafilters in terms of partitions. The combinatorial properties of these partition-ultrafilters, which we call Ramseyan ultrafilters, are similar to those of Ramsey ultrafilters. For example it will be shown that dual Mathias forcing restricted to a Ramseyan ultrafilter has the same features as Mathias forcing restricted to a Ramsey ultrafilter. Further we introduce an ordering on the set of partition-filters and consider the dual form of some cardinal characteristics of the continuum.


Introduction
The Stone-Čech compactification βN of the natural numbers, or equivalently, the ultrafilters over ω, is a well-studied space (cf. e.g. [vM90] and [CN74]) which has a lot of interesting topological and combinatorial features (cf. [HS98] and [To97]). In the late 1960's, a partial ordering on the non-principal ultrafilters βN \ N, the so-called Rudin-Keisler ordering, was established and "small" points with respect to this ordering were investigated rigorously (cf. [Bo70], [Bl73], [Bl81 1 ] and [La89]). The minimal points have a nice combinatorial characterization which is related to Ramsey's Theorem (cf. [Ra29,Theorem A]) and so, the ultrafilters which are minimal with respect to the Rudin-Keisler ordering are also called Ramsey ultrafilters (for further characterizations of Ramsey ultrafilters see [BJ95,Chapter 4.5]). Families, not necessarily filters, having similar combinatorial properties as Ramsey ultrafilters, are the so-called happy families (cf. [Ma77]), which are very important in the investigation of Mathias forcing (cf. [Ma77]).
From the category theoretical point of view, subsets of ω and partitions of ω are dual to each other (see e.g. [HLö∞ 1 , Introduction]), and therefore, it is natural to look for the dualization of statements about subsets of ω in terms of partitions of ω. In this dualization process, a lot of work is already done. Confer: [HLö∞ 1 ] for a dualization of βN; [CaSi84], [Ha98 1 ] and [HLö∞ 2 ] for the dualization of the Ramsey property and of Mathias forcing; [CaSi84] for a dualization of Ramsey's Theorem; [CKMW00] and [Ha98 2 ] for the dualization of some cardinal characteristics of the continuum.
To investigate partition-filters, a useful tool is missing: the dualization of Ramsey ultrafilters. The aim of this paper is to fill this gap.

Notations and definitions
Most of our set-theoretic notation is standard and can be found in textbooks like [Je78], [Ku83] or [BJ95]. So, we consider a natural number n as an ordinal, in particular n = {k : k < n} and 0 = ∅, and consequently, the set of natural numbers is denoted by ω. For a set S, P(S) denotes the power-set of S. The notation concerning partitions is not yet standardized. However, we will use the notation introduced in [Ha98 1 ].
A partition X of a set S consisting of pairwise disjoint, non-empty sets, such that X = S. The elements of a partition are called blocks. Mostly, we will consider partitions of ω, so, if not specified otherwise, the word "partition" refers to a partition of ω.
Most of the partitions in consideration are infinite, or in other words, contain infinitely many blocks. However, at some places we also have to consider finite partitions, this means, partitions containing only finitely many blocks. The unique partition containing just one block is denoted by {ω}. The set of all partitions is denoted by (ω) ≤ω and the set of all partitions containing infinitely many blocks is denoted by (ω) ω .
Let X and Y be two partitions of a set S. We say X is coarser than Y , or that Y is finer than X (and write X ⊑ Y ), if each block of X is the union of blocks of Y . Let X ⊓ Y denote the finest partition of S which is coarser than X and Y .
Further, for n ∈ ω and a partition X ∈ (ω) ≤ω , let X ⊓ {n} be the partition we get, if we glue all blocks of X together which contain a member of n. If X and Y are two partitions, then we write X ⊑ * Y if there is an n ∈ ω such that (X ⊓ {n}) ⊑ Y .
A set ⊆(ω) ≤ω is a partition-filter, if the following holds: properly contained in any partition-filter.
Notice that a partition-ultrafilter Í which does not contain a finite partition is always non-principal, and vice versa, a principal partition-ultrafilter always contains a finite partition, in fact it contains a 2-block partition (see [HLö∞ 1 , Fact 3.1]). Thus, ifÍ is a non-principal partition-ultrafilter, X ∈Í and X ⊑ * Y , then Y ∈Í .
In the sequel we are mostly interested in partition-filters which don't contain a finite partition, or in other words, in partition-filters ⊆(ω) ω .
For the sake of convenience, we defined the notion of partition-filter only for partition-filters over ω, but it is obvious how to generalize this notion for partitionfilters over arbitrary sets S (see also [HLö∞ 1 ]).

An ordering on the set of partition-filters
Let PF (ω) ≤ω denote the set of all partition-filters. We define a partial ordering on PF (ω) ≤ω which has some similarities with the Rudin-Keisler ordering on βN \ N.
Let f : ω։ω be any surjection from ω onto ω and let X ∈ (ω) ≤ω be any partition. Then f (X) denotes the finest partition such that whenever n and m lie in the same block of X, then f (n) and f (m) lie in the same block of f (X).
Since the identity map is a surjection and the composition of two surjections is again a surjection, the partial ordering " " is reflexive and transitive. Proof: Since is a partition-filter and f is a function, for any X 1 , X 2 ∈ we have X 1 ⊓ X 2 ∈ and , and therefore, À is a partition-filter. Further, The ordering " " induces in a natural way an equivalence relation "≃" on the set of partition-filters PF (ω) ≤ω : ≃ if and only if and .
So, the ordering " " induces a partial ordering of the set of equivalence classes of partition-filters. Concerning partition-ultrafilters, we get the following.
Fact 1.2.2 Let Í ,Î ∈ PUF (ω) ≤ω and assume that Í is principal or contains a partition, all of whose blocks are infinite. IfÍ ≃Î , then there is a permutation h of ω such that h(Í ) =Î .
Proof: Because Í Î and Î Í , there are surjections f and g from ω onto ω such that Î = f (Í ) and Í = g(Î ), and because Í and Î are both partitionultrafilters, by Fact 1.2.1 we get Í = f −1 (Î ) andÎ = g −1 (Í ). First assume that Í is principal and therefore contains a 2-block partition X = {b 0 , b 1 }. Because g −1 (X) ∈ Î , the partition-ultrafilter Î is also principal and we get Î = {Y ∈ (ω) ≤ω : ). If one of the blocks of X is finite, say b 0 , then f | b 0 as well as g| f (b 0 ) must be one-to-one, and therefore, b 0 has the same cardinality as c 0 . Hence, no matter if one of the blocks of X is finite or not, we can define a permutation h of ω such that h(b 0 ) = c 0 and h(b 1 ) = c 1 , which implies h(Í ) =Î . Now assume that Í contains a partition X = {b i : i ∈ ω}, all of whose blocks b i are infinite. Because g is a surjection, g −1 (X), which is a member of Î , is a partition, all of whose blocks are infinite. Let h be a permutation of ω such that The following proposition shows that " " is directed upward (for a similar result concerning the Rudin-Keisler ordering see [Bl73,p. 147]).
Proof: Let ρ 1 and ρ 2 be two functions from ω into ω defined by ρ 1 (n) := 2n and : n ∈ b}. Now, take any two partition-filters , ∈ PF (ω) ≤ω and define by Clearly, this defines a partition-filter. Define two surjections f and g from ω onto ω as follows: if n is even, 0 otherwise.
It is easy to verify that f ( ) = and g( ) = , which implies and . ⊣ 2 Ramseyan ultrafilters

Coloring segments
If X is a partition of a set S, then we say that S is the domain of X, written dom(X) = S. The set of all partitions of natural numbers n ∈ ω, called segments, is denoted by (N). Thus, s ∈ (N) implies dom(s) ∈ ω. In particular, ∅ is the unique partition of 0 and {{∅}} = {1} is the unique partition of 1. For s ∈ (N), |s| denotes the cardinality of s, which simply means the number of blocks of s, and s := {dom(s)}.
For a set b⊆ω, let min(b) be the least element of b and for a set P ⊆P(ω), let Min(P ) := {min(b) : b ∈ P }. Further, for a finite set b⊆ω, let max(b) be the greatest element of b. For X ∈ (ω) ≤ω , s ∈ (N) and n ∈ ω, let X(n) and s(n) be the nth block of X and s, respectively, where we start counting with 0 and assume that the blocks are ordered by their least element.
For n ∈ ω, (ω) n * denotes the set of all u ∈ (N) such that |u| = n. Further, for n ∈ ω and X ∈ (ω) ω let (X) n * := u ∈ (N) : |u| = n ∧ u * ⊑ X ; and if s ∈ (N) is such that |s| ≤ n and s ⊑ X, let From the so-called Dual Ramsey Theorem of Carlson and Simpson, which is Theorem 1.2 of [CaSi84], we get the following.
This combinatorial result is the dualization of Ramsey's Theorem, [Ra29, Theorem A], in terms of partitions.
We say that a surjection f : ω։ω respects the partition X ∈ (ω) ω , if we have f −1 (f (X)) = X, otherwise, we say that it disregards the partition X. If f −1 (f (X)) = {ω}, then we say that f completely disregards the partition X.
Lemma 2.1.2 For any surjection f : ω։ω and for any Z ∈ (ω) ω , there is an X ∈ (Z) ω such that f either respects or completely disregards the partition X.
Proof: Using Corollary 2.1.3 repeatedly, we can construct the partition X ∈ (ω) ω straight forward by induction on n. ⊣ We say that a family ⊆(ω) ω has the segment-coloring-property, if for every coloring of n∈ω (ω) (n+k+1) * with r + 1 colors, where r, k ∈ ω, and for any Z ∈ , there is an infinite partition X ∈ (Z) ω ∩ , such that for any n ∈ ω and for any s X with |s| = n + 1, (s, X) (n+k+1) * is monochromatic.
The next lemma shows that every partition-filter ∈ PF (ω) ω which has the segment-coloring-property is a partition-ultrafilter. A similar result we have for Ramsey filters over ω, since every Ramsey filter is an ultrafilter.
Proof: Let τ : [ω] n → r be any coloring of the n-element subsets of ω with r colors, where n and r are positive natural numbers. Define π : (ω) n * → r by stipulating π(s) := τ (Min(s * ) \ {0}). Take X ∈ Í such that (X) n * is monochromatic with respect to π, then, by the definition of π, the set [Min(X) \ {0}] n is monochromatic with respect to τ . ⊣ Ramsey ultrafilters over ω build the minimal points of the Rudin-Keisler ordering on βN \ N. This fact can also be expressed by saying that a non-principal ultrafilter U is a Ramsey ultrafilter if and only if any function g : ω → ω is either constant or one-to-one on some set of U. By Lemma 2.1.2, we get a similar result for Ramseyan ultrafilters with respect to the ordering " ".
Theorem 2.1.7 If Í is a Ramseyan ultrafilter, then for any surjection f : ω։ω there is an X ∈Í such that f either respects or completely disregards X.
Proof: The proof is the same as the proof of Lemma 2.1.2, but restricted to the partition-ultrafilter Í .

On the existence of Ramseyan ultrafilters
As we have seen in Lemma 2.1.6, every Ramseyan ultrafilter induces a Ramsey ultrafilter over ω. It is not clear if the converse holds as well. However, Ramseyan ultrafilters are always forceable: Let U ♭ be the forcing notion consisting of infinite partitions, stipulating X ≤ Y ⇔ X ⊑ * Y . U ♭ is the natural dualization of the forcing notion P(ω)/fin, ⊆ * , in the sequel denoted by U, and it is not hard to see that if is U ♭ -generic over V, then is a Ramseyan ultrafilter in V[ ]. Since U ♭ is σ-closed, as a consequence we get that Ramseyan ultrafilters exist if we assume the continuum hypothesis (denoted by CH). On the other hand we know by Lemma 2.1.6 that Ramseyan ultrafilters cannot exist if there are no Ramsey ultrafilters. Kenneth Kunen proved (cf. [Je78, Theorem 91]) that it is consistent with ZFC that Ramsey ultrafilters don't exist. We like to mention that Saharon Shelah showed that even p-points, which are weaker ultrafilters than Ramsey ultrafilters, may not exist (see [Sh98, VI §4]). He also proved that it is possible that-up to isomorphisms-there exists a unique Ramsey ultrafilter (see [Sh98, VI §5]).
In the following, c denotes the cardinality of the continuum and 2 c denotes the cardinality of its power-set. Andreas Blass proved that Martin's Axiom, denoted by MA, implies the existence of 2 c Ramsey ultrafilters (see [Bl73, Theorem 2]). He mentions in this paper that with CH in place of MA, this result is due to Keisler and with 1 in place of 2 c , it is due to Booth (cf. [Bo70, Theorem 4.14]). Further he mentions that his proof is essentially the union of Keisler's and Booth's proof. However, Blass' proof uses at a crucial point that MA implies that the tower number is equal to c. Such a result we don't have for partitions, because Timothy Carlson proved that the dual-tower number is equal to ℵ 1 (see [Mt86,Proposition 4.3]). So, concerning the existence of Ramseyan ultrafilters under MA, we cannot simply translate the proof of Blass, and it seems that MA and sets of partitions are quite unrelated. But as mentioned above, if one assumes CH, then Ramseyan ultrafilters exist. Moreover, with respect to the equivalence relation "≃" (defined in section 1.2) we get the following (for a similar result w.r.t. the Rudin-Keisler ordering see [Bl73, p. 149]).
Theorem 2.2.1 CH implies the existence of 2 c pairwise non-equivalent Ramseyan ultrafilters.
Proof: Assume V |= CH. Let χ be large enough such that P((ω) ω ) ∈ H(χ), i.e., the power set of (ω) ω (in V) is hereditarily of size < χ. Let N be an elementary submodel of H(χ), ∈ with |N| = ℵ 1 , containing all reals (or equivalently, all partitions) of V. We consider the forcing notion U ♭ in the model N.
. By construction, for any function ζ, the set G ζ := {X ∈ (ω) ω : X α ⊑ * X for some X α ∈ H ζ } is U ♭ -generic over N, thus, a Ramseyan ultrafilter in N[G ζ ], and since U ♭ is σ-closed and therefore adds no new reals, G ζ is also a Ramseyan ultrafilter in V. Furthermore, if ζ = ζ ′ , then the two Ramseyan ultrafilters G ζ and G ζ ′ are different (consider the two partitions X β+1 ∈ H ζ and X ′ β+1 ∈ H ζ ′ , where ζ(β) = ζ ′ (β)). Hence, in V, there are 2 c Ramseyan ultrafilters. Because there are only c surjections from ω onto ω, no equivalence class (w.r.t. "≃") can contain more than c Ramseyan ultrafilters, so, in V, there must be 2 c pairwise non-equivalent Ramseyan ultrafilters. ⊣ 3 The happy families' relatives 3.1 Relatively happy families As we will see below, the partition-families which have the segment-coloring-property are related to special coideals, so-called happy families, which are introduced and rigorously investigated by Adrian Mathias in [Ma77]. So, partition-families with the segment-coloring-property can be considered as "relatives of happy families".
Let us first consider the definition of Mathias' happy families.
Let [ω] ω be the set of all infinite subsets of ω, and let [ω] <ω be the set of all finite subsets of ω. A set I⊆P(ω) is a free ideal, if I is an ideal which contains the Fréchet ideal [ω] <ω . A set F ⊆P(ω) is a free filter, if {y : ω \ y ∈ F } is an ideal containing the Fréchet ideal. For a ∈ [ω] <ω , let a * := max{n + 1 : n ∈ a}, in particular, 0 * = 0. For x, y ∈ P(ω) we write y⊆ * x if (y \ x) ∈ [ω] <ω . For a set B⊆P(ω), let fil(B) be the free filter generated by B, so, x ∈ fil(B) if and only if there is a finite set y 0 , . . . , y n ∈ B such that (y 0 ∩ . . . ∩ y n )⊆ * x.
A set x⊆ω is said to diagonalize the family {x a : a ∈ [ω] <ω }, if x⊆x 0 and for all a ∈ [ω] <ω , if max(a) ∈ x, then (x \ a * )⊆x a .
The family A⊆P(ω) is happy, if P(ω) \ A is a free ideal and whenever fil{x a : a ∈ [ω] <ω }⊆A, there is an x ∈ A which diagonalizes {x a : a ∈ [ω] <ω }.
In terms of happy families one can define Ramsey ultrafilters as follows: A Ramsey ultrafilter is an ultrafilter that is also a happy family. Now we turn back to partitions. The Fréchet ideal corresponds to the set of finite partitions, and therefore, the notion of a free filter corresponds to partition-filters containing only infinite partitions, hence, to partition-filters ⊆(ω) ω . For a set ⊆(ω) ω , let fil( ) be the partition-filter generated by , so, X ∈ fil( ) if and only if there is a finite set of partitions Y 0 , . . . , Y n ∈ such that (Y 0 ⊓. . .⊓Y n ) ⊑ * X.
A partition X is said to diagonalize the family {X s : s ∈ (N)}, if X ⊑ X ∅ and for all s ∈ (N), if s * X, then s * ⊓ X ⊑ X s .
An example of a relatively happy family is (ω) ω , the set of all infinite partitions (compare with [Ma77, Example 0.2]). Another example of a much smaller relatively happy family is given in the following theorem (compare with [Ma77, p. 63]).
Theorem 3.1.1 Every Ramseyan ultrafilter is relatively happy.
So, we are always in the former case, which completes the proof. ⊣

A game characterization
There is a characterization of happy ultrafilters over ω, i.e., of Ramsey ultrafilters, in terms of games (cf. [BJ95, Theorem 4.5.3]). A similar characterization we get for relatively happy partition-ultrafilter.
Let Í be a partition-ultrafilter. Define a game G(Í ) played by players I and II as follows: Player I on the n-th move plays a partition X n ∈ Í . Player II responds with a segment s n ∈ (N) such that |s n | = n, s * n−1 s n and for all m < n, s * m ⊓ s * n ⊑ X m+1 , where s 0 := ∅. Player I wins if and only if the unique partition X with s n X (for all n) is not inÍ .
Theorem 3.2.1 Let Í ∈ PUF (ω) ω , then player I has a winning strategy in G(Í ) if and only if Í is not relatively happy.
Proof: Assume first that the partition-ultrafilter Í is relatively happy and that {X s : s ∈ (N)} is a strategy for player I. This means, player I begins with X ∅ and then, if s n is the n-th move of player II, player I plays X sn . BecauseÍ is relatively happy, there is a partition X ∈ Í which diagonalizes the family {X s : s ∈ (N)}, in particular, X ⊑ X ∅ . Now, by the definition of X and by the rules of the game G(Í ), player II can play the segments of X. More precisely, player II plays on the n-th move the segment s n , so that |s n | = n and s * n X. Since X ∈Í , the strategy {X s : s ∈ (N)} was not a winning strategy for player I. Now assume that the strategy σ = {X s : s ∈ (N)} is not a winning strategy for player I. Consider the game where player I is playing according to the strategy σ. In this game, player II can play segments s n such that the unique partition X with s n X (for all n) is in Í . We have to show that X diagonalizes the family {X s : s ∈ (N)}. For n ∈ ω, let s n ∈ (N) be such that s * n X and |s n | = n. Fix m ∈ ω, then, by the rules of the game, for any n > m we have s * m ⊓ s * n ⊑ X m+1 , which implies s * m ⊓ X ⊑ X m+1 . Since player I follows the strategy σ, X m+1 = X sm , and because m was arbitrary, for all m ∈ ω we get s * m ⊓ X ⊑ X sm . Hence, X diagonalizes the family {X s : s ∈ (N)}. ⊣ Mathias forcing restricted to a non-principal ultrafilter U, denoted by M U , consists of the ordered pairs a, x ∈ M, where in addition we require that x ∈ U.
Mathias forcing has a lot of nice combinatorial properties (some of them are mentioned below) which also hold for Mathias forcing restricted to a Ramsey ultrafilter (see [Ma77]).
The dual Ellentuck topology on (ω) ω is the topology induced by the dual Ellentuck neighborhoods (defined in section 2.1). Now, the dual Mathias forcing M ♭ , introduced in [CaSi84], is defined similarly to Mathias forcing M, using the dual Ellentuck topology instead of the Ellentuck topology. So, M ♭ consists of ordered pairs s, X such that (s, X) ω is a dual Ellentuck neighborhood and the ordering on whereǓ is the canonical U ♭ -name for the U ♭ -generic object (U ♭ as in section 2.2).
(ii) Pure decision: For any M-condition a, x and any sentence Φ of the forcing language M, there is an M-condition a, y ≤ a, x such that either a, y M Φ or a, y M ¬Φ. For any M ♭ -condition s, X and any sentence Φ of the forcing language M ♭ , there is an M ♭ -condition s, Y ≤ s, X such that either s, Y M ♭ Φ or s, Y M ♭ ¬Φ. (iii) Homogeneity property: If x G is M-generic over V and y ∈ [x G ] ω , then y is also M-generic over V.
If X G is M ♭ -generic over V and Y ∈ (X G ) ω , then Y is also M ♭ -generic over V.
In [Ha98 1 ] it is shown that if ⊆(ω) ω is a so-called game-family, then M ♭ has pure decision and the homogeneity property ([Ha98 1 , Thm. 4.3 & 4.4]). Gamefamilies have the segment-coloring-property and therefore, the so-called game-filters, i.e., game-families which are partition-filters, are Ramseyan ultrafilters. Unlike for Ramseyan ultrafilters, it is not clear if CH implies the existence of game-filters, so, it seems that game-filters are stronger than Ramseyan ultrafilters. However, in the sequel we show that if Í ∈ PUF (ω) ω is a Ramseyan ultrafilter, then M ♭ Í has pure decision and the homogeneity property.
Recently, Stevo Todorčević gave an abstract presentation of Ellentuck's theorem by introducing the notion of a quasi ordering with approximations which admits a finitization and the notion of a Ramsey space. The Abstract Ellentuck Theorem says that a quasi ordering with approximations which admits a finitization and satisfies certain axioms is a Ramsey space.
LetÍ ∈ PUF (ω) ω be a partition-ultrafilter and let "⊑" be the quasi ordering on Í . For each n ∈ ω, let the function p n :Í → (N) be such that p n (X) is the unique s with s * X and |s| = n. Let p be the sequence (p n ) n∈ω . It is easy to verify that the triple (Í , ⊑, p) is a quasi ordering with approximations. For n, m ∈ ω and X, Y ∈Í define: p n (X) ⊑ fin p m (Y ) if and only if dom p n (X) = dom p m (Y ) and p n (X) ⊑ p m (Y ). This definition verifies that (Í , ⊑, p) admits a finitization. If (s, X) ω is a dual Ellentuck neighborhood and X ∈Í , then (s, X) ω ∩Í is called a Í -dual Ellentuck neighborhood. The topology on Í , induced by the Í -dual Ellentuck neighborhoods, is called theÍ -dual Ellentuck topology. With respect to the Í -dual Ellentuck topology, the topological space Í is a Ramsey space, if for any subset S⊆Í which has the Baire property with respect to the Í -dual Ellentuck topology, and for any Í -dual Ellentuck neighborhood (s, Y ) ω ∩Í , there is a partition X ∈ (s, Y ) ω ∩Í such that either (s, X) ω ∩Í ⊆S or (s, X) ω ∩Í ⊆Í \S.
Moreover, we get the following two results. Proof: Let Φ be any sentence of the forcing language M ♭ Í . With respect to Φ we characteristics, as well as for the relation between them, we refer the reader to [Va90].
First we consider the shattering cardinal h. This cardinal was introduced in [BPS80] as the minimal height of a tree π-base of βN \ N. Later it was shown by Szymon Plewik in ( [Pl86]) that h = add(r 0 ) = cov(r 0 ), where r 0 denotes the ideal of Ramsey-null sets. It is easy to see that p ≤ h, and therefore, MA(σ-centered) implies h = c.
The dual form of the classical cardinal characteristics were introduced and investigated in [CKMW00] and further investigated in [Ha98 2 ]. Concerning the dualshattering cardinal H, one easily gets ℵ 1 ≤ H ≤ h, and in [Ha98 2 ] it is shown that H > ℵ 1 is consistent relative to ZFC and that H = add(R 0 ) = cov(R 0 ), where R 0 denotes the ideal of dual Ramsey-null sets. After all these symmetries, one would not expect the following: MA + (c > H) is consistent relative to ZFC. This was proved by Jörg Brendle in [Br00] and implies that H < p is consistent relative to ZFC.
Concerning the reaping and the dual-reaping number r and R, respectively, the situation looks different. It is shown in [Ha98 1 ] that p ≤ R ≤ min{r, i}, and thus we get MA(σ-centered) implies R = c. Further, it is easy to show that R ≤ U, where U denotes the partition-ultrafilter base number, i.e., the dual form of u, and consequently, MA(σ-centered) implies U = c.
For a Ramsey ultrafilter U, Brendle introduced in [Br95] the ideal r 0 U , which is the ideal of Ramsey-null sets with respect to the ultrafilter U. Concerning this ideal r 0 U , he showed for example that hom ≤ non(r 0 U ), where hom is the homogeneity number investigated by Blass in [Bl93, Section 6]. There, Blass also investigated the so-called partition number par and showed that par = min{b, s}. Now, replacing the Ramsey ultrafilter U by a Ramseyan ultrafilterÍ , one obtains the ideal R 0 Í of dual Ramseynull sets with respect to Í as the dualization of the ideal r 0 U , and replacing the colorings of [ω] 2 -involved in the definition of hom and par-by colorings of (ω) 2 * , one obtains the cardinal characteristics Hom and Par and could begin to investigate them. But this is left to the reader. ❦