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Parametrizing the abstract Ellentuck theorem

We give a parametrization with perfect sets of the abstract Ellentuck theorem. The main tool for achieving this goal is a sort of parametrization of an abstract version of the Nash-Williams theorem. As corollaries, we obtain some known classical results like the parametrized version of the Galvin-Prikry theorem due to Miller and Todorcevic, and the parametrized version of Ellentuck's theorem due to Pawlikowski. Also, we obtain parametized vesions of nonclassical results such as Milliken's theorem.


Introduction.
In [3], Ellentuck considers the space N [∞] , of all the infinite sets of natural numbers, with the exponential (or Ellentuck's) topology to obtain his famous topological generalization of Ramsey's theorem [14]. The basic open sets of this topology are the neighborhoods [a, α] = {β ∈ N [∞] : a ⊂ β ⊆ α}, where a (resp. α) is a finite (resp. infinite) subset of N. We recall that a set X ⊆ N [∞] is Ramsey (completely Ramsey in [3]) if for every Ellentuck neighborhood [a, α] there exists an infinite β ⊆ α such that either [a, β] ⊆ X or [a, β] ∩ X = ∅. In [3], Ellentuck proved that a set is Ramsey if and only if it has the Baire property relative to Ellentuck's topology, proving in this way that the family of Ramsey sets is closed under the Souslin operation and hence obtaining a simpler topological proof of Silver's theorem [15] that classical analytic subsets of N [∞] are Ramsey, and generalizing the Galvin-Prikry theorem [6] that classical Borel subsets of N [∞] are Ramsey. Theorems analog to that of Ellentuck have also been proven on other spaces and different contexts, provided an analog "exponential" topology is given in each case (see for instance [2], [4] and [10]). These are the so called Ellentuck type theorems. Spaces where an Ellectuck type theorem can be proven are called (topological) Ramsey spaces. The main tool for proving such theorems (including Ellentuck's) is deviced for each particular case as a sort of "combinatorial forcing", inspired by the works of Galvin-Prikry [6] and Nash-Williams [12]. Recently, based on a previous work of Carlson and Simpson (see [1]), S. Todorcevic has isolated abstract combinatorial features which are conditions of sufficiency for topological spaces provided of a suitable "exponential" topology, to guarantee that Ellentuck type theorems can be obtained in such spaces. These combinatorial features enable to device a sort of "abstract Galvin-Prikry machinery" which constitutes the main means for obtaining an abstract Ellectuck theorem. It turns out that spaces satisfying these conditions are Ramsey spaces. We present a summary of Todorcevic's recent presentation of the abstract topological Ramsey theory in section 2.
In this work we will show that the conditions proposed in [16], and summarized in the next section, are also sufficient to obtain a parametrized version of the abstract Ellentuck theorem. In this way, we obtain as corollaries some known classical results like the parametrized version of the Galvin-Prikry theorem due to Miller and Todorcevic [9], and the parametrized version of the Ellentuck theorem due to Pawlikowski [13] which makes use of the notion of abstract Baire property of Morgan [11]. Also, we obtain parametized vesions of nonclassical results such as Milliken's theorem [10], and we prove that the family of perfectly Ramsey  k , of infinite block basic sequences of functions p : N → {0, 1, · · · , k} of finite domain and with p(n) = k for some n, will be described in section 4.

Abstract Topological Ramsey Theory. Summary of main facts.
All the definitions and results throughout this section were taken from [17] and are expected to appear in [16]. A previous presentation of the following notions can also be found in [1].
We will consider triplets of the form (R, ≤, (p n ) n∈N ), where R is a set, ≤ is a quasi order on R, N is the set of natural numbers, and for every n ∈ N, p n : R → P n is a function with range P n . For each A ∈ R, we say that p n (A) is the nth approximation of A. In order to capture the combinatiorial structure required to ensure the provability of an Ellentuck type theorem, we will impose some assumptions on (R, ≤, (p n ) n ). The first three of them are the following: These three assumptions allow us to identify each A ∈ R with the sequence (p n (A)) n of its approximations. In this way, if we consider the space P = n P n with the discrete topology, we can identify R with a subspace of the (metric) space P [∞] (with the product topology) of all the sequences of elements of P. Via this identification, we will regard R as a subspace of P [∞] , and we will say that R is metrically closed if it is a closed subspace of P [∞] .
Also, for a ∈ P we define the lenth of a, |a|, as the unique n such that a = p n (A) for some A ∈ R.
We also consider on R the Ellentuck type neighborhoods where a ∈ P and A ∈ R. If [a, A] = ∅ we will say that a is compatible with A (or A is compatible with a). Let P[A] = {a ∈ P : a is compatible with A}.
We write [n, A] for [p n (A), A], and Exp(R) for the family of all the neighborhoods [n, A]. This family generates the natural "exponential" topology on R which is finer than the product topology.
A sequence ([n k , A k ]) k of elements of Exp(R) is called a fusion sequence if it is infinite and if (i) (n k ) k ⊆ N is nondecreasing and lim k→∞ n k = ∞, Since R is closed, building fusion sequences constitute a very useful procedure for defining desired elements of R: for every fusion sequence ([n k , A k ]) k we have that k [n k , A k ] = ∅ (and a singleton). The limit of the fusion sequence is the unique element A ∞ of k [n k , A k ]. Note that p n k (A ∞ ) = p n k (A k ), for all k.
With this notation, we can define an analog notion for subsets of R, to that of Ramseyness for subsets of N In [16] it is shown that A1, A2 and A3, together with the following three assumptions are conditions of suficiency for a triplet (R, ≤, (p n ) n ), with R metrically closed, to be a Ramsey space (see also [1]): (A4)(Finitization) There is a quasi order ≤ f in on P such that: Given a and A, we define the depth of a in A, depth A (a), as the minimal n such that a ≤ f in p n (A).
(A5) (Amalgamation) Given compatible a and A with depth A (a) = n, the following holds: . With these definitions, A1-A6 are easily verified. In this case A6 reduces to a natural variation of the classical pigeon hole principle for finite partitions of an infinite set of natural numbers. Note also that N [∞] is easily identified with a closed subspace of P [∞] , namely, the set of all the sequences (x n ) n of finite sets such that x n = x n+1 \{max(x n+1 )}, for each n ∈ N. Then (N [∞] , ⊆, (p n ) n )) is a Ramsey space in virtue of the abstract Ellentuck theorem. Hence, Ellentuck's theorem is obtained as corollary: , the following hold: (a) X is Ramsey iff X has the Baire Property, relative to Ellentuck's topology. (b) X is Ramsey null iff X is meager, relative to Ellentuck's topology.

Parametrizing with perfect sets.
In this section we will present our main result. We recall that 2 ∞ denotes the space of infinite sequences of 0's and 1's, with the product topology regarding 2 = {0, 1} as a discrete space. Also, 2 <∞ denotes the set of finite sequences of 0's and 1's. Let us consider some features of the perfect subsets of 2 ∞ , following [13]: where the first ramification occurs. Then, set (u * i)(Q) = σ * i, i = 0, 1. Here " * " denotes "concatenation". Note that for each n, Q = {Q(u) : u ∈ 2 n }.
Given n ∈ N, and perfect sets S and Q we say that S ⊆ n Q if S(u) ⊆ Q(u), for every u ∈ 2 n . The relation "⊆ n " is a partial order. If for every u ∈ 2 n we have chosen a S u ⊆ Q(u), then S = u S u is perfect and we have S(u) = S u and S ⊆ n Q. As pointed out in [13], the most important feature of this partial order is the property of fusion: if Q n+1 ⊆ n+1 Q n , n ∈ N, then the fusion Q = n Q n is a perfect set and Q ⊆ n Q n , for each n.
Our goal in this section is proving a parametrized version of the abstract Ellentuck theorem of the previous section. This constitutes the main result contained in this work and is stated as thereom 3.0.3 below.
We introduce now the abstract version of the notion of "perfectly Ramsey" (see [13]). We will use the same name: given a triplet (R, ≤ , (p n ) n ) as defined at the previous section, we say that a set X ⊆ 2 ∞ × R is perfectly Ramsey if for every perfect set Q ⊆ 2 ∞ and every neighborhood [a, A] = ∅ there exist a perfect set Also, we will need to generalize the notion of abstract Baire property (see [11]) to this context: Let P be the family of perfect subsets of 2 ∞ . We will say that a set X ⊆ 2 ∞ × R has the P × Exp(R)-Baire property if for every perfect set Q ⊆ 2 ∞ and every neighborhood [a, A] there exist a perfect set The main tool for proving theorem 3.0.3 is the following fact which is a sort of parametrization of an abstract version of Nash-Williams' theorem: Theorem 3.0.4. Given (R, ≤, (p n ) n ), with R metrically closed and satisfying A1-A6, the following is true: For every F ⊆ 2 <∞ × P, perfect P ⊆ 2 ∞ and A ∈ R there exist a perfect S ⊆ P and D ≤ A such that one of the following holds:   [4]). Before proving it, we need to start up our parametrized combinatorial machinery, based on the tecniques used in [4], [13] [13] and [17].
Combinatorial Forcing 1. Fix F ⊆ 2 <∞ × P. For a perfect Q ⊆ 2 ∞ , A ∈ R and a pair (u, a) ∈ 2 <∞ ×P, we say that (Q, A) accepts (u, a) if for every x ∈ Q(u) and for every B ∈ [a, A] there exist integers k and m such that (x| k , p m (B)) ∈ F . We say that (Q, A) rejects (u, a) if for every perfect S ⊆ Q(u) and every B ≤ A, compatible with a, (S, B) does not accepts (u, a). Also, we say that (Q, A) decides (u, a) if it accepts or rejects it.
Combinatorial Forcing 2. Fix X ⊆ 2 ∞ ×R. For a perfect Q ⊆ 2 ∞ , A ∈ R and a pair (u, a) ∈ 2 <∞ × P, we say that (Q, A) accepts (u, a) if Q(u) × [a, A] ⊆ X . We say that (Q, A) rejects (u, a) if for every perfect S ⊆ Q(u) and every B ≤ A, compatible with a, (S, B) does not accepts (u, a). And as before, we say that (Q, A) decides (u, a) if it accepts or rejects it.
Suppose we have defined Q n and B n such that (Q n , B n )decides every (u, a) ∈ 2 n × P[B n ] with depth Bn (a) = n. Let u 0 , u 1 ,... , u 2 n+1 −1 be a list of the elements of 2 n+1 ; and let b 0 , b 1 ,... , b r be a list of the b ∈ P[B n ] such that depth Bn (b) = n + 1.
In a similar way, for every (i, j) ∈ {0, 1, · · · , 2 n+1 −1} ×{0, 1, · · · , r}, we can find Q i,j n and B i,j n with: n ]; and such that (Q i,j n , B i,j n ) decides (u i , b j ). (Notice that this construction is possible in virtue of A5(i). Again, we can assume B i,j n ∈ [n + 1, B n ] in virtue of A5(ii) and lemma 3.0.5(b)). Let Besides, Q n+1 ⊆ n+1 Q n and B n+1 ∈ [n + 1, In the same way, for every (i, j) ∈ {0, 1, · · · , 2 n+1 −1}×{0, 1, · · · , r}, we can find a D i,j n with: Now take D ∈ n [n, D n ]. Then D is as required.  3. (a) The implication from left to right is obvious. So suppose X ⊆ 2 ∞ × R has the P × Exp(R)-Baire Property, and let P ×[a, A] be given. In order to make the proof notationally simpler, we will assume a = ∅ without a loss of generality. Claim 3.0.8. GivenX ⊆ 2 ∞ × R, perfectP ⊆ 2 ∞ andÂ ∈ R, there exist Q ⊆P and B ≤Â such that for each (u, b) ∈ 2 <∞ × P[B] with |u| ≥ depth B (b) one of the following holds: Proof of theorem Claim. Consider the Combinatorial Forcing 2 and apply lemma 3.0.6.
Apply the claim to X , P and A to find Q 1 ⊆ P and B 1 ≤ A such that for each (u, b) ∈ 2 <∞ × P[B 1 ] with |u| ≥ depth B 1 (b) one of the following holds: ( Apply the claim to X c , S 1 and D 1 to find Q 2 ⊆ S 1 and B 2 ≤ D 1 such that for each (u, b) ∈ 2 <∞ × P[B 2 ] with |u| ≥ depth D 2 (b) one of the following holds: As before, for each t ∈ T Q 2 , choose u t 2 ∈ 2 <∞ such that u t 2 (Q 2 ) ⊑ t. Let Notice that for every (t, b) ∈ T S 2 × P[D 2 ] the following holds: ]∩X c we reach to a similar contradiction in virtue of (ii) and (4) above. So there is neither But this is impossible because X has the P × Exp(R)-Baire Property.
(b) Again, the implication from left to right is obvious. Conversely, the result follows easily from (a) and the fact that X is P × Exp(R)meager.
This completes the proof of theorem 3.0.3.

Some Particular Cases.
Several interesting consequences can be derived from the facts obtained in the previous section. Some of them are known classical results and the others are parametrized versions of known Ellentuck type theorems in nonclassical spaces.
A block basic sequence is any finite or infinite sequence X = (x n ) n∈I⊆N of elements of F IN k such that max(supp(x n )) < min(supp(x m )) whenever n < m.
We shall use a, b, c, ... for finite block basic sequences, and A, B, C, ... for infinite block basic sequences. In this latter case we will assume that the set of indexes is I = N.
In [16] this is called the tetris operation. For every j ∈ N, T (j) is the j-th iteration of T , where T (0) (p) = p.
For a given block basic sequence X = (x n ) n∈I⊆N , the subspace of F IN k generated by X, denoted by [X], is the set of elements of F IN k of the form: where n 0 < n 1 < · · · < n r is a finite sequence of elements of I and j 0 < j 1 < · · · < j r is a sequence of elements of {0, 1, · · · , k} such that j i = 0 for some i ∈ {0, 1, · · · , r}.
The following result shows us an important feature of F IN k , which provides us of a pigeon hole principle within this context: Theorem 4.0.9. [Gowers [5]] For every integer r > 0 and every par- In the case k = 1, F IN k is the set F IN of nonempty finite subsets of N, and theorem 4.0.9 is Hindman's theorem [7].
be the set of infinite block basic sequences and define, k . Also, for every A ∈ F IN [∞] k , let the n-th approximation of A be p n (A) = the first n elements of A.
Then the set P of approximations is F IN Then Q×[0, B]∩ n X n = ∅: take (x, C) ∈ Q×[0, B] and fix arbitrary n. To show that (x, C) ∈ X n let k be large enough so that depth B (p k (C)) = m ≥ n. Then by construction Q×[p k (C), B] ∩X m = ∅ and hence, since X n ⊆ X m , we have (x, C) ∈ X n . This completes the proof. Now, we borrow some terminology from [13]: Let A be a family of subsets of a set Z. We say that X , Y ⊆ Z are compatible (with respect to A) if there exists W ∈ A such that W ⊆ X ∩ Y. Also, we say that A is M-like if for any B ⊆ A such that |B| < |A|, every member of A which is not compatible with any member of B is compatible with Z \ B.
Notice that the family P of perfect subsets of 2 ∞ is M-like, as well as the family Exp(R) (this is true of any topological basis). Therefore, according to lemma 2.7 in [13], if we require that |Exp(R)| = |P| (= 2 ℵ 0 ), then the family P × Exp(R) = {P × [n, A] : P ∈ P and A ∈ R} is also M-like. This lead us to the following: Corollary 5.0.13. Let (R, ≤, (p n ) n∈N ) satisfying A1-A6, with R metrically closed be such that |R| = 2 ℵ 0 . Then, the family of perfectly Ramsey subsets of 2 ∞ × R is closed under the Souslin operation.
Proof. In virtue of theorem 3.0.3, the family of perfectly Ramsey subsets of 2 ∞ ×R coincides with the family of subsets of 2 ∞ ×R which have the P × Exp(R)-Baire property. And as we pointed out in the previous parragraph, P × Exp(R) is M-like. So the proof follows from lemma 5.0.12on top of this section and lemmas 2.5 and 2.6 of [13] (which refer to a well-known result of Marczewski [8]).