Ben Gurion University and Carnegie Mellon University RULES AND REALS

MARTIN GOLDSTERN AND MENACHEM KOJMAN

Abstract. A “k-rule” is a sequence A⃗ = ((A_{n}, B_{n}) : n < ℕ) of pairwise disjoint sets B_{n}, each of cardinality ≤ k and subsets A_{n} ⊆ B_{n}. A subset X ⊆ ℕ (a “real”) follows a rule A⃗ if for infinitely many n ∈ ℕ, X ∩ B_{n} = A_{n}.

Two obvious cardinal invariants arise from this definition: the least number of reals needed to follow all k-rules, s_{k}, and the least number of k-rules with no real that follows all of them, r_{k}.

Call A⃗ a bounded rule if A⃗ is a k-rule for some k. Let r_{∞} be the least cardinality of a set of bounded rules with no real following all rules in the set.

We prove the following: r_{∞} ≥ max(cov(𝕂), cov(𝕃)) and r = r_{1} ≥ r_{2} = r_{k} for all k ≥ 2. However, in the Laver model, r_{2} < b = r_{1}.

An application of r_{∞} is in Section 3: we show that below r_{∞} one can find proper extensions of dense independent families which preserve a pre-assigned group of automorphisms. The original motivation for discovering rules was an attempt to construct a maximal homogeneous family over ω. The consistency of such a family is still open.

Introduction

In the present paper we present new cardinal invariants which emerged in the authors’ investigations of homogeneous families. These numbers have intrinsic interest for the theory of the continuum (in fact we regard it as surprising that those numbers have not been discovered earlier).

In Section 1 we discuss cardinal invariants related to “k-rules.” A k-rule is a sequence A⃗ = ((A_{n}, B_{n}) : n < ω) of pairwise disjoint sets B_{n}, each of cardinality ≤ k, and subsets A_{n} ⊆ B_{n}. A subset X ⊆ ω (a “real”) follows a rule A⃗ if for infinitely many n ∈ ω, X ∩ B_{n} = A_{n}.

A rule A⃗ is bounded if it is a k-rule for some k ∈ ω.

The obvious cardinal invariants related to rules are the following: the least number of reals needed to follow all k-rules, s_{k}, and the least number of k-rules with no real following all of them, r_{k}. Let r_{∞} be the least number of bounded rules with no real following all of them.

We compare the r_{k}s and r_{∞} among themselves and to well known cardinal invariants: covering of category, covering of Lebesgue measure, r, b, d and the evasion numbers e_{k} which were studied by Blass and Brendle. We prove:

(a) max(cov(𝕂), cov(𝕃)) ≤ r_{∞};

(b) r = r_{1} ≥ r_{2} = r_{k} for all k ≥ 2;

(c) s_{2} ≤ e_{2};

(d) r_{∞} ≤ min(r_{2}, d).

In Section 2 we prove the consistency of r_{2} < b.

In Section 3 we show that below r_{∞} one can properly extend an independent family of subsets of ω preserving a prescribed group of automorphisms. This is the relevance of r_{∞} to the behavior of homogeneous families under inclusion, which was the original motivation for the discovery of rules.

1. Rules

Definition 1.1. (1) A rule is a sequence A⃗ = (A_{n}, B_{n} : n ∈ ω), where the sets B_{n}

are disjoint and finite, and for all n, A_{n} ⊆ B_{n} ⊆ ω.

(2) We say that X ∈ [ω]^{ω} follows the rule A⃗ if there are infinitely many n with X ∩ B_{n} = A_{n}; otherwise X is said to avoid A⃗.

(3) For k ∈ ω we say that A⃗ is a k-rule if all sets B_{n} have size ≤ k. We say that A⃗ is a bounded rule if A⃗ is a k-rule for some k.

(4) More generally, for any function f : ω → ω we say that A⃗ is an f-rule if for all n, |B_{n}| ≤ f(n). We say that f is a “slow” function if

∑_{n=0}^{∞} 2^{−f(n)} = ∞,

and we say that A⃗ is a slow rule if it is an f-rule for some slow f. Definition 1.2. (1) For k ∈ ω let r_{k} := min{|R| : there is no X which follows all

k-rules from R }. (Similarly r_{f}, when f : ω → ω.)

(2) Dually, let s_{k} := min{|S| : every k-rule is followed by some X ∈ S }.

(3) We let r_{∞} = min{|R| : there is no X which follows all bounded rules from R}.

1

Date: January 1997.

The first author is supported by the Austrian Science Foundation (FWF).

The second author was partially supported by an NSF grant no. 9622579.

We remark that 2^{k} trivially bounds the least cardinality of a set of k-rules with the property that every real follows some rule in the set.

Recall that the “splitting” number s and the “reaping” number r are defined as follows:

Definition 1.3. If s, X ∈ [ω]^{ω}, then we say that s “splits” X if s divides X into two infinite parts, i.e., s ∩ X and (ω − s) ∩ X are both infinite.

(1) s := min{|S| : S ⊆ [ω]^{ω}, every X ∈ [ω]^{ω} is split by some s ∈ S }

(2) r := min{|R|: R ⊆ [ω]^{ω}, there is no X ∈ [ω]^{ω} which splits all r ∈ R }

Fact 1.4. · · · ≤ r_{3} ≤ r_{2} ≤ r_{1} = r, and s ≤ s_{2} ≤ s_{3} ≤ · · ·. However, s_{1} = 2, witnessed by S = {∅, ω}.

Theorem 1.5. (a) Let (N, ∈) be a model of ZFC* (a large enough fragment of ZFC). If a real X follows all rules from N, then X is Cohen over N. (Conversely, a Cohen real over N follows all rules from N.)

(b) If X is random over N, then X follows all slow rules from N (so in particular, all bounded rules).

(c) max(cov(𝕂), cov(𝕃)) ≤ r_{∞}. (cov(𝕂) is the smallest number of first category sets needed to cover the real line. cov(𝕃) is defined similarly using measure zero sets.)

Proof. (a): Assume that X ⊆ ω follows all rules from N. We claim that χ_{X}, the characteristic function of X, is a Cohen real over N, that is, the set {χ_{X}↾n : n ∈ ω} is generic for the forcing notion ^{<ω}2.

To verify this claim, consider any nowhere dense tree T ⊆ ^{<ω}2 in N. We have to check that χ_{X} is not a branch of T.

Using the fact that T is nowhere dense (and T is in N) we can by induction (in N!) find sequences (n_{i} : i < ω) and (η_{i} : i < ω) such that for all i < ω we have:

1. n_{i} < n_{i+1}, η_{i} ∈ ^{[n_{i},n_{i+1})}2

2. For all ν ∈ ^{n_{i}}2, ν ∪ η_{i} ∉ T.

Now let B_{i} := [n_{i}, n_{i+1}), A_{i} = {k : η_{i}(k) = 1}. Our assumption tells us that X follows the rule (A_{i}, B_{i} : i ∈ ω). So for some i we have X ∩ B_{i} = A_{i}, and hence χ_{X} ⊇ η_{i}. Hence χ_{X} is not a branch of T.

This concludes the proof of (a).

The converse to (a) is obvious.

(b) is also easy: Let X_{n} := {X : X ∩ B_{n} ≠ A_{n}}. For n ≠ m, the sets X_{n} and X_{m} are independent (in the probabilistic sense), and μ(X_{n}) = 1 − 2^{−f(n)}, where μ is the Lebesgue measure on P(ω) ≃ ^{ω}2. Hence μ(⋂_{n>m} X_{n}) = ∏_{n>m}(1 − 2^{−f(n)}) = 0.

(c) follows from (a) and (b).

Theorem 1.6 (Shelah). For k ≥ 2, r_{k} = r_{2} (and similarly, s_{2} = s_{k}).

Proof. We will show that r_{k} = r_{k+1}: Let N_{0} be sufficiently closed (say, a model of ZFC*, but closed under some recursive functions is sufficient) of size < r_{k}; we have to show that there is a real that follows all k + 1-rules from N_{0}.

We define a sequence (N_{i}, C_{i} : i ≤ k) such that N_{i} ∪ {C_{i}} ⊆ N_{i+1}, each N_{i} is sufficiently closed and of the same cardinality as N_{0}, and C_{i} follows all k-rules from N_{i}.

Now let C be the “average” of the C_{i}: m ∈ C iff m is in “most” of the C_{i}’s, or formally:

C := {m ∈ ω : |{i ≤ k : m ∈ C_{i}}| > (k + 1)/2}

Now we check that C indeed follows all k + 1-rules from N_{0}.

Let (A_{n}, B_{n} : n ∈ ω) be a k + 1-rule in N_{0}. For 0 ≤ i ≤ k we let (A_{n}^{i}, B_{n}^{i} : n ∈ ω) be the k-rule obtained by removing the each ith element of B_{n}. That is, letting {b_{n}^{0}, …, b_{n}^{k}} be the increasing enumeration of B_{n} we let B_{n}^{i} := B_{n} \ {b_{n}^{i}}, A_{n}^{i} := A_{n} ∩ B_{n}^{i}.

Let E_{0} := ω. For 0 ≤ i ≤ k let

E_{i+1} := {n ∈ E_{i} : B_{n}^{i} ∩ C_{i} = A_{n}^{i}},

i.e., E_{i+1} is the set of indices on which C_{i} follows the rule (A_{n}^{i}, B_{n}^{i} : n ∈ E_{i}). Note that E_{i} ∈ N_{i} and C_{i} ∈ N_{i+1}. By the choice of C_{i} we know that each E_{i+1} is infinite.

We conclude the proof by showing that for n ∈ E_{k+1} we have A_{n} = B_{n} ∩ C. Let n ∈ E_{k+1} (so also n ∈ E_{i} for all i ≤ k), and m ∈ B_{n}. Say m = b_{n}^{j}. Then for i ≠ j we have m ∈ B_{n}^{i}, so m ∈ A_{n}^{i} ⇔ m ∈ C_{i}.

Hence the cardinality of the set {i ≤ k : m ∈ C_{i}} is either in {0, 1} (iff m ∉ A_{n}) or in {k, k + 1}. In any case we get m ∈ C iff m ∈ A_{n}. So A_{n} = B_{n} ∩ C.

Theorem 1.7. r_{∞} ≥ min(r_{2}, d). In particular, if r_{2} ≤ d then r_{∞} = r_{2}.

Proof. Let N be a model of ZFC* of cardinality < min(r_{2}, d). We will find a real X following all bounded rules from N.

Define sequences (N_{i} : i < ω), (X_{i} : i < ω) satisfying the following conditions:

1. N_{0} = N.

2. N_{i} is a model of ZFC*, N_{i−1} ∪ {X_{i}} ⊆ N_{i}.

3. |N_{i}| = |N_{0}|.

4. X_{i} follows all i-rules (and hence also all j-rules for j ≤ i) from N_{i−1}. Let N_{ω} be a model of size |N_{0}| containing (N_{i} : i < ω) and (X_{i} : i < ω). Since |N_{ω}| < d we can find a strictly increasing function f that is not dominated by any function from N_{ω}.

Define X ⊆ ω by requiring X ∩ (f(i − 1), f(i)] = X_{i} ∩ (f(i − 1), f(i)]. We claim that X follows all bounded rules from N.

To complete the proof, consider an arbitrary k-rule (A_{n}, B_{n} : n ∈ ω) from N. We may assume min ⋃_{n} B_{n} > f(k). We define sequences (E_{i} : k ≤ i < ω) satisfying the following conditions for all i ≥ k.

1. ∀n ∈ E_{i} B_{n} ∩ X_{i} = A_{n}.

2. E_{i} ∈ N_{i}.

3. E_{i+1} ⊆ E_{i}. We can carry out this construction, because (A_{n}, B_{n} : n ∈ E_{i}) is a rule in N_{i}, so we just choose E_{i+1} to witness that X_{i+1} follows this rule.

Now let n_{i} := min E_{i}. Clearly the function i ↦ n_{i} is in N_{ω}. So we can find infinitely many j such that f(j) > max B_{n_{j}}.

We claim that for each such j, X ∩ B_{n_{j}} = A_{n_{j}}. For all i ∈ [k, j] we have n_{j} ∈ E_{i}, so X_{i} ∩ B_{j} = A_{j}. Note that B_{j} ⊆ [f(k), f(j)), so we also have X ∩ B_{j} = A_{j}.

Problem 1.8. Is r_{∞} < r_{2} consistent?

We remark that in the random real model we have r_{2} = cov(𝕃) = c = r_{∞}, d = ℵ_{1}. So one cannot hope to prove r_{2} ≤ d.

We now consider the invariant that is dual to r_{k}, and we compare it with the well-known “evasion” number.

Definition 1.9. (π, D) is a k-predictor, if D is an infinite subset of ω, π = (π_{n} : n ∈ D), π_{n} a function from ^{n}k to k.

We say that f ∈ ^{ω}k evades (π, D) if there are infinitely many ℓ ∈ D such that f(ℓ) ≠ π_{ℓ}(f↾ℓ).

e_{k} := min{|N| : ∀π ∃f ∈ N : f evades π}

Brendle in [1] investigated these and other cardinal invariants and showed that all e_{k} are equal to each other.

The following construction connects rules with predictors.

Definition 1.10. Let R = (A_{n}, B_{n} : n ∈ ω) be a 2-rule. Define a 2-predictor (π_{R}, D_{R}) as follows:

1. D_{R} = {max B_{n} : n ∈ ω}

2. If ℓ = max B_{n}, and |A_{n}| = 1, then π_{ℓ}(f) = f(min B_{n}) for all f ∈ ^{ℓ}2. Otherwise, π_{ℓ}(f) = 1 − f(min B_{n}).

Lemma 1.11. Let X ⊆ ω. If χ_{X} evades π_{R}, then either X or ω \ X follows R.

Proof. Let ℓ_{n} := max B_{n}, i_{n} = min(B_{n}) for all n.

X evades π_{R}, so there are infinitely many n such that X(ℓ_{n}) ≠ π_{ℓ_{n}}(X↾ℓ_{n}).

Case 1: There are infinitely many such n where in addition |A_{n}| = 1.

So for each such n, X(ℓ_{n}) ≠ π_{ℓ_{n}}(X↾ℓ_{n}) = X(i_{n}). So X(ℓ_{n}) ≠ X(i_{n}), so X ∩ B_{n} must be either A_{n} or B_{n} \ A_{n}. One of the two alternatives holds infinitely often. Hence, either there are infinitely many n such that X ∩ B_{n} = A_{n}, or there are infinitely many n such that (ω \ X) ∩ B_{n} = A_{n}.

Case 2: There are infinitely many such n with X(ℓ_{n}) ≠ π_{ℓ_{n}}(X↾ℓ_{n}), where in addition |A_{n}| = 2, i.e., A_{n} = B_{n}. So for each such n, X(ℓ_{n}) ≠ π_{ℓ_{n}}(X↾ℓ_{n}) = 1 − X(i_{n}). So X(ℓ_{n}) = X(i_{n}), so X ∩B_{n} must be either B_{n} or ∅. One of the two alternatives holds infinitely often. So again we either get infinitely many n such that X ∩ B_{n} = A_{n}, or infinitely many n such that (ω \ X) ∩ B_{n} = A_{n}.

Case 3: For infinitely many n as above we have A_{n} = ∅. Similar to the above.

Corollary 1.12. s_{2} ≤ e_{2}

Proof. Let N be a model (of set theory) witnessing e_{2}, i.e., for every 2-predictor π there is a function f ∈ N evading π.

Let R be any 2-rule. There is X ∈ N evading π_{R}, so either X or ω \ X (both in N) follows R.

Remark 1.13. s ≤ e_{2} is known. Brendle showed that s < s_{2} is consistent (unpublished).

2. Consistency of r_{2} < r

We show here in contrast to theorem 1.6 that r is not provably equal to r_{2}. Moreover, whereas b ≤ r is provable in ZFC (see [6] for a collection of results on cardinal invariants), we show that r_{2} < b is consistent with ZFC.

The following definition is standard:

Definition 2.1. 1. S is a slalom iff dom(S) = ω and for all n ∈ ω, S(n) is a

finite set of size n.

2. If f is a function with dom(f) = ω, S a slalom, then we say that S captures f iff ∀^{∞}n f(n) ∈ S(n).

3. Let M ⊆ N be sets (typically: models of ZFC*). We say that N has the Laver

property over M iff:

For every function H ∈ ^{ω}ω∩M, for every function f ∈ ^{ω}ω∩N satisfying f ≤ H there is a slalom S ∈ M that captures f.

4. A forcing notion P has the Laver property iff ⊩_{P} “V^{P} has the Laver property over V.”

Before we formulate the main lemma, we need the following easy claim:

Claim 2.2. Let k > 2^{n}. If X ⊆ ^{k}2, |X| = n then there are i < j in k such that for all f ∈ X, f(i) = f(j).

Proof. For i < j, f ∈ X, define an equivalence relation ∼_{f} by: i ∼_{f} j ⟺ f(i) = f(j). Let i ∼ j iff i ∼_{f} j for all f in X. Since each ∼_{f} has at most 2 equivalence classes, ∼ has at most 2^{n} classes, so there are i ≠ j, i ∼ j.

Lemma 2.3. Assume that (N, ∈) is a model of ZFC*, and that V has the Laver property over N.

Then every real avoids some 2-rule from N.

Proof. Let a_{0} = 0, a_{n+1} = a_{n} + 2^{n} + 1. The sequence (a_{n} : n ∈ ω) is in N. For any X ∈ P(ω), we will find a rule in N which X does not follow. Let χ_{X} ∈ ^{ω}2 be the characteristic function of X. Define X^{∗} := (χ_{X}↾[a_{n}, a_{n+1}) :

n ∈ ω). Note that there are only 2^{2^{n}+1} many possibilities for χ_{X}↾[a_{n}, a_{n+1}).

Since V has the Laver property over N there is a sequence S⃗ = (S_{n} : n ∈ ω) ∈ N, S_{n} ⊆ ^{[a_{n},a_{n+1})}2, |S_{n}| ≤ n, and for all n > 0, χ_{X}↾[a_{n}, a_{n+1}) ∈ S(n). By the above claim we can find i_{n} < j_{n} in [a_{n}, a_{n+1}) such that for all z ∈ S(n), z(i_{n}) = z(j_{n}). Since the sequence S⃗ is in N, we can find such a sequence (i_{n}, j_{n} : n < ω) in N.

Define a 2-rule (A_{n}, B_{n} : n ∈ ω) ∈ N by A_{n} = {i_{n}}, B_{n} = {i_{n}, j_{n}}. Since i_{n} ∈ X iff j_{n} ∈ X, X does not follow this rule.

Lemma 2.4. (a) Let P̄ = (P_{i}, Q_{i} : i < ω_{2}) be a countable support iteration of proper forcing notions such that for each i we have ⊩_{i} “Q_{i} has the Laver property.” Then P_{ω_{2}}, the countable support limit of P̄, also has the Laver property.

(b) Laver forcing is proper and has the Laver property.

(c) Laver forcing adds a real that dominates all reals from the ground model.

Proof. These facts are well known and (at least for the case where each Q_{i} is Laver forcing) appear implicitly or explicitly in Laver’s paper [4].

Conclusion 2.5. Let P_{ω_{2}} be the limit of a countable support iteration of Laver forcing over a model V_{0} of GCH. Then ⊩_{P_{ω_{2}}} b = r = ω_{2} and r_{2} = ω_{1}.

Proof. Let V_{ω_{2}} = V^{P_{ω_{2}}}. V_{ω_{2}} ⊧ b = ω_{2} is well known. (Let f_{i} be the real added by the ith Laver forcing; then (f_{i} : i < ω_{2}) is a strictly increasing and cofinal sequence in ^{ω}ω.

By 2.4, V_{ω_{2}} has the Laver property over V_{0}. Hence, by 2.3, every real avoids some rule from V_{0}. So r_{2} ≤ |^{ω}ω ∩ V_{0}| = ℵ_{1}.

3. Application to independent families

A family F ⊆ P(ω) of subsets of ω is independent iff it generates a free boolean algebra in P(ω)/fin. Equivalently, for any two disjoint finite subsets of F, the intersection of all members in the first set with all complements of members in the second set is infinite.

The following is an example of an independent family of size continuum over a countable set: {A_{r} : r ∈ ℝ} where A_{r} = {p ∈ ℤ[X] : p(r) > 0}.

A family F ⊆ P(ω) is dense iff for any two finite disjoint subsets of ω there are infinitely many members of F that contain the first set and are disjoint to the second.

An interesting (proper) subclass of the class of dense independent families over ω is the class of homogeneous families, which was introduced in [2]. Its study was continued in [3].

While every dense independent family is contained in a maximal dense independent family, this is not obvious (and perhaps false) for homogeneous families. The existence, even the consistency, of a maximal homogeneous family over ω is still open. In particular, an increasing union of homogeneous families need not be homogeneous.

In the study of extendibility of homogeneous families, the following notion is fundamental: Let G ⊆ Aut F. We define (F, G) ≤ (F′, G′) iff F ⊆ F′, G ⊆ G′ ⊆ Aut F′. The usefulness of ≤ is that unions of suitable ≤ chains are homogeneous (see [3] for a detailed account of direct limits in the category of homogeneous families).

We show now that below r_{∞} one can get proper ≤-extensions of independent families. This was our original motivation for discovering r_{∞}.

Theorem 3.1. Suppose G ⊆ Aut F, F ⊆ P(ω) is dense independent and |F| + |G| < r_{∞}. Then there exists F′ ⫌ F such that (F, G) ≤ (F′, G)

Proof. Suppose that G ⊆ Aut F, F is dense independent and |G| + |F| < r_{∞}. We shall find a real X ⊆ ω such that X ∉ F and F ∪ G[X] is independent, where G[X] is the orbit of X under G. This will suffice, since clearly G ⊆ Aut(F ∪ G[X]) for any real X.

It is a priori unclear why such X should exist. If for example there is some σ ∈ G with finite support, then for no X ⊆ ω is even the orbit G[X] itself independent. However, the following lemma takes care of this. Let supp(σ) = {n ∈ ω : σ(n) ≠ n} for a permutation σ ∈ Sym ω.

Lemma 3.2. Suppose that F is dense independent and σ ∈ Aut F is not the identity. Then there are distinct sets C_{n} ∈ F such that for all n, C_{2n} − C_{2n+1} ⊆ supp σ.

Remark 3.3. In particular, the supports of non-identity automorphisms have the finite intersection property and hence generate a filter. This is the “strong Mekler condition” for Aut F (see [5]).

Proof. Fix k^{∗} ∈ ω for which σ(k^{∗}) ≠ k^{∗}. Find C_{2n}, C_{2n+1} by induction on n. Suppose C_{m} is chosen for m < 2n. By density, there are infinitely many C ∈ F for which k^{∗} ∈ C, σ(k^{∗}) ∉ C. Choose some such C so that neither C nor σ[C] are among {C_{m} : m < 2n}. Let C_{2n} = C and C_{2n+1} = σ[C]. Since k^{∗} ∈ C_{2n} − C_{2n+1}, those sets are indeed distinct.

We claim that C_{2n} − C_{2n+1} ⊆ supp(σ). Indeed, for any k ∈ C_{2n} − C_{2n+1} we have σ(k) ∈ C_{2n+1} but k ∉ C_{2n+1}, so k ≠ σ(k).

We now continue the proof of theorem 3.1. Let M be a transitive model of ZFC* of cardinality < r_{∞} such that F, G ∈ M and G ⊆ M, F ⊆ M. Let X be a real that follows all bounded rules from M. Clearly, X ∉ M, and therefore X ∉ F.

We need to show that every boolean combination over F ∪ G[X] is infinite. Suppose that

D := A ∩ σ_{0}[X] ∩ ⋯ ∩ σ_{n−1}[X] ∩ (ω − σ_{n}[X]) ∩ ⋯ ∩ (ω − σ_{m−1}[X])

is a boolean combination over F ∪ G[X], where σ_{i} ∈ G for i < m, and A is some boolean combination over F. Clearly, A ∈ M.

Set N = \binom{m}{2} and let (τ_{i} : i < N) be a list of all σ_{k} ∘ σ_{ℓ}^{−1} for k < ℓ < m. By induction find a sequence (C_{0}, …, C_{2n+1}) of 2N + 2 many distinct sets such that no C_{k} participates in D and such that C_{2k} − C_{2k+1} ⊆ supp τ_{k}. C_{2k} and C_{2k+1} are constructed in the k-th step by using lemma 3.2.

Since all the C_{k} are distinct, the intersection E = A ∩ ⋂_{k<N} C_{2k} − ⋃_{k<N} C_{2k+1} is infinite. Clearly E belongs to M. Define by induction an m-rule (A_{n}, B_{n} : n < ω) as follows: suppose (A_{k}, B_{k} : k ≤ n) are defined. Find a point j_{n} ∈ E such that B = {σ_{k}^{−1}(j_{n}) : k < m} is disjoint from ⋃_{ℓ≤n} B_{ℓ} and j_{n} ∉ ⋃_{ℓ≤n} B_{ℓ}. Let B_{n+1} be B and let A_{n+1} = {σ_{ℓ}^{−1}(j_{n}) : ℓ < n}.

The rule we defined obviously belongs to M. Since X satisfies all bounded rules from M, there are infinitely many n for which X ∩ B_{n} = A_{n}. For each such n, X_{n} ∈ D.

References

[1] Jörg Brendle. Evasion and prediction—the Specker phenomenon and Gross spaces. Forum Math., 7(5):513–541, 1995.

[2] Martin Goldstern, Rami Grossberg, and Menachem Kojman. Infinite Homogeneous Bipartite Graphs With Unequal Sides. Discrete Mathematics, 149:69–82, 1996.

[3] Menachem Kojman and Saharon Shelah. Homogeneous families and their automorphism groups. Journal of the London Mathematical Society, 52:303–317, 1995.

[4] R. Laver. On the consistency of Borel’s conjecture. Acta Math., 137:151–169, 1976.

[5] J. K. Truss. Embeddings of infinite permutation groups. In Proceedings of groups — St Andrews 1985, volume 121 of London Math. Soc. Lecture Series, pages 335–351. Cambridge University Press, 1986.

[6] J.E. Vaughan. Small uncountable cardinals in topology. In Open problems in topology, pages 217–218. Elsvier Science Publishers, B.V. North Holland, 1990. ed. van Mill, J. and Reed, G.M.