Categoricity, Amalgamation, and Tameness

Theorem. For each 2 ≤ k < ω there is an Lω1,ω-sentence φk such that: (1) φk is categorical in μ if μ ≤ אk−2; (2) φk is not אk−2-Galois stable; (3) φk is not categorical in any μ with μ > אk−2; (4) φk has the disjoint amalgamation property; (5) For k > 2, (a) φk is (א0,אk−3)-tame; indeed, syntactic first-order types determine Galois types over models of cardinality at most אk−3; (b) φk is אm-Galois stable for m ≤ k − 3; (c) φk is not (אk−3,אk−2)-tame. We adapt an example of [9]. The amalgamation, tameness, stability results, and the contrast between syntactic and Galois types are new; the categoricity results refine the earlier work of Hart and Shelah and answer a question posed by Shelah in [17]. Considerable work (e.g. [14, 15, 16, 7, 8, 6, 18, 12, 11]) has explored the extension of Morley’s categoricity theorem to infinitary contexts. While the analysis in [14, 15] applies only to Lω1,ω, it can be generalized and in some ways strengthened in the context of abstract elementary classes. Various locality properties of syntactic types do not generalize in general to Galois types (defined as orbits under an automorphism group) in an AEC [5]; much of the difficulty of the work stems from this difference. One such locality properties is called tameness. Roughly speaking, K is (μ, κ)-tame if distinct Galois types over models of size κ have distinct restrictions to some submodel of size μ. For classes with arbitrarily large models, that satisfy amalgamation and tameness, strong categoricity transfer theorems have been proved [7, 8, 6, 13, 4, 10]. In particular these results yield categoricity in every uncountable power for a tame AEC in a countable language (with arbitrarily large models satisfying amalgamation and the joint embedding property) that is categorical in any single cardinal above א2 ([6]) or even above א1 ([13]). In contrast, Shelah’s original work [14, 15] showed (under weak GCH) that categoricity up to אω of a sentence in Lω1,ω implies categoricity in all uncountable cardinalities. Hart and Shelah [9] showed the necessity of the assumption by constructing sentences φk which were categorical up to some אn but not eventually November 1, 2007, The first author is partially supported by NSF grant DMS-0500841. 1 2 JOHN T. BALDWIN AND ALEXEI KOLESNIKOV categorical. These examples were thus a natural location to look for examples of categoricity and failure of tameness. The example expounded here is patterned on the one in Hart-Shelah, [9]: our analysis of their example led to the discovery of some minor inaccuracies (the greatest categoricity cardinal is אk−2 rather than אk−1). Although the properties we assert could be proved with more complication for the original example, we present a simpler example. In Section 1 we describe the example and define the sentences φk. In Section 2 we introduce the notion of a solution and prove lemmas about the amalgamation of solutions. From these we deduce in Section 5 positive results about tameness. In some sense, the key insight of this paper is that the amalgamation property holds in all cardinalities (Section 3) while the amalgamation of solutions is very cardinal dependent. We prove in Section 4 that this example is a model-complete AEC. We show in Section 6 that φk is not Galois stable in אk−2 and deduce the non-tameness. From the instability we derive in Section 7 the failure of categoricity in all larger cardinals, thus answering the question posed by Shelah as Problem 6.12 in [17]. Baldwin and Shelah [5] showed under often satisfied conditions (K admits intersections i.e. is closed under arbitrary intersections) amalgamation does not affect tameness. That is, for any tameness spectrum realized by an AEC K which admits intersections, there is another which has the amalgamation property but the same tameness spectrum. But this construction destroys categoricity so those examples do not address the weaker conjecture that the amalgamation property together with categoricity in a finite number of cardinals implies (א0,∞)-tameness. We refute that conjecture here. Baldwin, Kueker and VanDieren [2] showed that if K is an (א0,∞)-tame AEC with arbitrarily large models that is Galois-stable in κ it is Galois stable in κ+; our results show the tameness hypothesis was essential. This paper and [5] provide the first examples of AEC that are not tame. In both papers the examples are built from abelian groups. But while [5] obtains non-tameness from phenomena that are closely related to the Whitehead conjecture and so to non-continuity results in the construction of groups, this paper shows the failure can arise from simpler considerations. 1. THE BASIC STRUCTURE This example is a descendent of the example in [3] of an א1-categorical theory which is not almost strongly minimal. That is, the universe is not in the algebraic closure of a strongly minimal set. Here is a simple way to describe such a model. Let G be a strongly minimal group and let π map X onto G. Add to the language a binary function t : G × X → X for the fixed-point free action of G on π−1(g) for each g ∈ G. That is, we represent π−1(g) as {ga : g ∈ G} for some a with π(a) = g. Recall that a strongly minimal group is abelian and so this action of G is strictly 1-transitive. This guarantees that each fiber has the same CATEGORICITY, AMALGAMATION, AND TAMENESS 3 cardinality as G and π guarantees the number of fibers is the same as |G|. Since there is no interaction among the fibers, categoricity in all uncountable powers is easy to check. Let k ≥ 2 be a natural number. Notation 1.1. The formal language for this example contains unary predicates I,K,G,G∗, H,H∗; a binary function eG taking G × K to H; a function πG mapping G∗ to K, a function πH mapping H∗ to K, a 4-ary relation tG on K × G × G∗ × G∗, a 4-ary relation tH on K × H × H∗ × H∗. Certain other projection functions are in the language but not expressly described. These symbols form a vocabulary L′; we form the vocabulary L by adding a (k + 1)-ary relation Q on (G∗)k ×H∗. We start by describing the L′-structure M(I) constructed from any set I with at least k elements. Typically, the set I will be infinite; but it is useful to have all the finite structures as well. We will see that the L′-structure is completely determined by the cardinality of I . So we need to work harder to get failure of categoricity, and this will be the role of the predicate Q. The structure M(I) is a disjoint union of sets I,K,H,G,G∗ and H∗. Let K = [I] be the set of k-element subsets of I . H is a single copy of Z2. Let G be the direct sum of K copies of Z2. So G, K, and I have the same cardinality. We include K, G, and Z2 as sorts of the structure with the evaluation function eG: for γ ∈ G and k ∈ K, eG(γ, k) = γ(k) ∈ Z2. So in Lω1,ω we can say that the predicate G denotes exactly the set of elements with finite support of Z2. Now, we introduce the sets G∗ and H∗. The set G∗ is the set of affine copies of G indexed by K. First, we have a projection function πG from G∗ onto K. Thus, for u ∈ K, we can represent an element x of π−1 G (u) in the form (u, x′) ∈ G∗. Alternatively, we say that x ∈ Gu. We refer to the set π−1 G (u) as the G∗stalk, or fiber over u. Then we encode the affine action by the relation tG ⊂ K×G×G∗×G∗ which is the graph of a regular transitive action ofG onGu. That is, for all x = (u, x′), y = (u, y′) there is a unique γ ∈ G such that tG(u, γ, x, y) holds. (Of course, this can be expressed in Lω,ω.) As a set, H∗ = K × Z2. As before if πH(x) = v holds x has the form (v, x′), and we denote by H∗ v the preimage π −1 H (v). Finally, for each v ∈ K, tH ⊂ K × Z2 ×H∗ ×H∗ is the graph of a regular transitive action of Z2 on the stalk H∗ v . (∗): We use additive notation for the action of G (H) on the stalks of G∗ (of H∗). (1) For γ ∈ G, denote the action by y = x+ γ whenever it is clear that x and y come from the same G∗-stalk. It is also convenient to denote by y − x the unique element γ ∈ G such that y = γ + x. 4 JOHN T. BALDWIN AND ALEXEI KOLESNIKOV (2) For δ ∈ H , denote the action by y = x+ δ, whenever it is clear that x and y come from the same H∗-stalk. Say that δ = y − x. If I is countably infinite, let ψ1 k be the Scott sentence for the countably infinite L′-structure M(I) based on I that we have described so far. This much of the structure is clearly categorical (and homogeneous). Indeed, suppose two such models have been built on I and I ′ of the same cardinality. Take any bijection between I and I ′. To extend the map to G∗ and H∗, fix one element in each partition class (stalk) in each model. The natural correspondence (linking those selected in corresponding classes) extends to an isomorphism. Thus we may work with a canonical L′-model; namely with the model that has copies of G (without the group structure) as the stalks Gu and copies of Z2 (also without the group structure) as the stalks H∗ v . The functions tG and tH impose an affine structure on the stalks. Notation 1.2. The L-structure is imposed by a (k+ 1)-ary relation Q on (G∗)k × H∗, which has a local character. We will use only the following list of properties of Q, which are easily axiomatized in Lω1,ω: (1) Q is symmetric, with respect to all permutations, for the k elements from G∗; (2) Q((u1, x1), . . . , (uk, xk), (uk+1, xk+1)) implies that u1, . . . , uk+1 form all the k element subsets of a k + 1 element subset of I . We call u1, . . . , uk+1 a compatible (k + 1)-tuple; (3) using the notation introduced at (*) Q is related to the actions tG and tH as follows: (a) for all γ ∈ G, δ ∈ H Q((u1, x1), . . . , (uk, xk), (uk+1, xk+1)) ⇔ ¬Q((u1, x1 + γ), . . . , (uk, xk), (uk+1, xk+1)) if and only if γ(uk+1) = 1; (b) Q((u1, x1), . . . , (uk, xk), (uk+1, xk+1)) ⇔ ¬Q((u1, x1), . . . , (uk, xk), (uk+1, xk+1 + δ)) if and only if δ = 1. Let ψ2 k be the conjunction of sentences expressing (1)–(3) above, and we let φk := ψ1 k ∧ ψ2 k. It remains to show that such an expansion to L = L′ ∪ {Q} exists. We do this by explicitly showing how to define Q on the canonical L′-structure. In fact, we describe 2|I|·|K| such structures parameterized by functions `. Fact 1.3. Let M = M(I) be an L′-structure described above. Let K := [I]k. Let ` : I ×K → 2 be an arbitrary function. CATEGORICITY, AMALGAMATION, AND TAMENESS 5 For each compatible (k+1)-tuple u1, . . . , uk+1, such that u1∪· · ·∪uk+1 = {a} ∪ uk+1 for some a ∈ I and uk+1 ∈ K, define an expansion of M to L by M |= Q((u1, x1), . . . , (uk, xk), (uk+1, xk+1)) if and only if x1(uk+1) + · · · + xk(uk+1) + xk+1 = `(a, uk+1) mod 2. Then M satisfies the properties (1)–(3) of Notation 1.2. Indeed, it is straightforward to check that the expanded structure M satisfies the properties. We describe the interaction of G and Q a bit more fully. Using symmetry in the first k components, we obtain the following property that was used by Hart and Shelah to define Q in [9]. Fact 1.4. For all γ1, . . . , γk ∈ G and all δ ∈ H we have Q((u1, x1), . . . , (uk, xk), (uk+1, xk+1)) ⇔ Q((u1, x1 + γ1), . . . , (uk, xk + γk), (uk+1, xk+1 + δ)) if and only if γ1(uk+1) + · · ·+ γk(uk+1) + δ = 0 mod 2. In order to consider finite L-structures with L′-reducts of the form M(I) for some of our inductive proofs, we introduce the following terminology. Definition 1.5. We call an L-structure N a full structure for φk if N L′ is isomorphic to an M(I) for some I and N |= ψ2 k. Let χk be the disjunction of the sentences describing M(I) for each finite set I . Let φ̂k be φk ∨ (ψ2 k ∧ χk). Then we can write “the L-structure N is a full structure for φk” more shortly as N |= φ̂k. An L-substructure A of M |= φk is called a full substructure if A |= φ̂k. Remark 1.6. (1) For infinite N , full structure is the same as being a model of φk; φ̂k includes structures built on a finite I . (2) The need for the notion of a full substructure can be explained, for example, by the fact that a subset {a0, a1, a2} of I(M) together with a single element x ∈ Ga0,a1 is a substructure, but not a full substructure, of M |= φ2. We want to close such a substructure under almost all the Skolem functions, excluding the ones that add elements of the “spine” I . In the next section, we show that φk is categorical in א0, . . . ,אk−2. So in particular φk is a complete sentence for all k. (See Chapter 7 of [1] for an account of completeness of sentences in Lω1,ω.) 6 JOHN T. BALDWIN AND ALEXEI KOLESNIKOV Now we obtain abstract elementary classes (Kk,≺K) where Kk is the class of models of φk and for M,N |= φk, M ≺K N if M ≺Lω1,ω N . We show in Section 4 that M ⊂ N implies M ≺Lω1,ω N for models of φk. We freely use various notions from the general theory of AEC, such as Galois type, below. All are defined in [1]. For convenience we repeat the three most used definitions. Definition 1.7. The AEC K has the disjoint amalgamation property if for any M0 ≺ M1,M2, there is a model M |= φk with M M0 and embeddings fi : Mi → M , i = 1, 2 such that f1(M1) ∩ f2(M2) = f1(M0) = f2(M0). If we omit the requirement on the intersection of the images, we have the amalgamation property. Under assumption of amalgamation (disjointness is not needed) and joint embedding one can construct monster models, i.e., strongly model homogeneous models M of an appropriate large size. (See [1] for the definitions and the construction.) Joint embedding is clear in our context and we prove amalgamation in Section 3. Using monster models, one can give the following simple definition of a Galois type. Definition 1.8. Let K be an AEC with amalgamation. Let M ∈ K, M ≺K M and a ∈ M. The Galois type of a over M (∈ M) is the orbit of a under the automorphisms of M which fix M . The set of all Galois types over M is denoted ga-S(M). In a class with amalgamation we can check whether two points have the same Galois type by the following criterion: For M ≺K N1 ∈K, M ≺K N2 ∈ K and a ∈ N1 −M , b ∈ N2 −M , the Galois type a over M in N1 is the same as the Galois type b over M in N2 if there exist strong embeddings f1, f2 of N1, N2 into some N∗ which agree on M and with f1(a) = f2(b). Definition 1.9. We say K is ω-Galois stable if for any countable M ∈ K, |ga-S(M)| = א0. Definition 1.10. We say K is (χ, μ)-tame if for any N ∈K with |N | = μ, for all p, q ∈ ga-S(N), if p N0 = q N0 for every N0 ≤ N with |N0| ≤ χ, then p = q. 2. SOLUTIONS AND CATEGORICITY As we saw in Fact 1.3, the predicate Q can be defined in somewhat arbitrary way. Showing categoricity of the L-structure amounts to showing that any model M , of an appropriate cardinality, is isomorphic to the model where all the values of ` are chosen to be zero; we call such a model a standard model. This motivates the following definition: CATEGORICITY, AMALGAMATION, AND TAMENESS 7 Definition 2.1. Fix a model or a full structure M . A solution for M is a selector f that chooses (in a compatible way) one element of the fiber in G∗ above each element of K and one element of the fiber in H∗ above each element of K. Formally, f is a pair of functions (g, h), where g : K(M) → G∗(M) and h : K(M) → H∗(M) such that πGg and πHh are the identity and for each compatible (k + 1)-tuple u1, . . . , uk+1: Q(g(u1), . . . , g(uk), h(uk+1)). Notation 2.2. As usual k = {0, 1, . . . k − 1} and we write [A]k for the set of k-element subsets of A. We will show momentarily that if M and N have the same cardinality and have solutions fM and fN then M ∼= N . Thus, in order to establish categoricity of φk in א0, . . . ,אk−2, it suffices to find a solution in an arbitrary model of φk of cardinality up to אk−2. Our approach is to build up the solutions in stages, for which we need to describe selectors over subsets of I(M) (or of K(M)) rather than all of I(M). Definition 2.3. We say that (g, h) is a solution for the subset W of K(M) if for each u ∈ W there are g(u) ∈ Gu and h(u) ∈ H∗ u such that if u1, . . . , uk, uk+1 are a compatible (k + 1)-tuple from W , then Q(g(u1), . . . , g(uk), h(uk+1)). If (g, h) is a solution for the setW , whereW = [A]k for someA ⊂ I(M), we say that (g, h) is a solution over A. Remark 2.4. Let k ≥ 2, and let M be a model of φ̂k. If A ⊂ I(M) has k elements, then there is a solution over A. Indeed, [A]k is a singleton, so there are no restrictions coming from the predicate Q. Definition 2.5. The models of φk have the extension property for solutions over sets of size λ (or over finite sets) if for every M |= φk, any solution (g, h) over a set A with |A| = λ (or A finite), and every a ∈ I(M) − A there is a solution (g′, h′) over the set A ∪ {a}, extending (g, h). One can treat the element g(u) as the image of the element (u, 0) under the isomorphism between the standard model andM , where 0 represents the constantly zero function in the stalk Gu. Not surprisingly, we have the following: Lemma 2.6. If M and N are models of φk of the same cardinality and have solutions fM and fN then M ∼= N . Moreover, suppose K has solutions and has extension of solutions for models of cardinality less than |M |. If g is an isomorphism between full substructures (or submodels) M ′, N ′ of M and N with |M ′| < |M | and |N ′| < |N |, then the isomorphism ĝ between M and N can be chosen to extend g. Finally, if 8 JOHN T. BALDWIN AND ALEXEI KOLESNIKOV fM ′ is a solution on M ′ which extends to a solution fM on M , then ĝ maps them to a similar extending pair on N ′ and N . Proof. We prove the ‘moreover’ clause; the first statement is a special case when g is empty and the ‘finally’ is included in the proof. Say, g maps M ′ to N ′. Without loss of generality,M L′ = M(I),N L′ = M(I ′). Let α be a bijection between I and I ′ which extends g I . Extend naturally to a map from K(M) to K(N) and from G(M) to G(N), which extends g on M ′. By assumption there is a solution fM ′ on M ′. It is clear that g maps fM ′ to a solution fN ′ on N ′; by assumption fN ′ extends to a solution on N . (Note that if we do not have to worry about g, we let α be an arbitrary bijection from I to I ′ and let α(fM (u)) be fN (α(u)).) For x ∈ G∗(M −M ′) such that M |= πG(x) = u, there is a unique a ∈ G(M) with a = x − fM (u) (the operation makes sense because a and fM (u) are in the same stalk). Let α(x) be the unique y ∈ N −N ′ such that N |= tG(α(u), α(a), fN (α(u)), y) i.e., y = α(a) + fN (α(u)) in the stalk Gα(u)(N). Do a similar construction for H∗ and observe that Q is preserved. 2.6 We temporarily specialize to the case k = 2. Claim 2.7. The models of φ̂2 have the extension property for solutions over finite sets. Proof. Let A := {a0, . . . , an−1}, let (g, h) be a solution over A, and suppose a is not in A. For each v = {a, ai}, let yv be an arbitrary element of H∗ v . Now extend h to the function h′ with domain [A ∪ {a}]2 by defining h′(v) := yv. It remains to define the function g′ on each {a, ai}, and we do it by induction on i. For i = 0, pick an arbitrary starting point1 x ∈ Ga,a0 . Let γ0 ∈ G be such that for j = 1, . . . , n− 1: γ0(a, aj) = 1 if and only if M |= ¬Q(({a, a0}, x), g(a0, aj), h′(a, aj)). It is clear that γ ∈ G(M) and that letting g′({a, a0}) := ({a, a0}, x+γ0), we have a partial solution. 1For an argument in Section 4, we will need to choose this point more carefully; we will use the term “starting point” then. CATEGORICITY, AMALGAMATION, AND TAMENESS 9 Suppose that g′({a, aj}), j < i, have been defined. Pick an arbitrary starting point x ∈ Ga,ai . Let γi ∈ G(M) be such that for j ∈ {0, . . . , n− 1} \ {i} γi(a, aj) = 1 if and only if M |= ¬Q(({a, ai}, x), g(ai, aj), h′(a, aj)). Also let γ′ i ∈ G(M) be such that for j < i γ′ i(ai, aj) = 1 if and only if M |= ¬Q(({a, aj}, x), g′(a, aj), h(ai, aj)). Now letting g′({a, ai}) := ({a, ai}, x+ γi + γ′ i) yields a well-defined solution on A ∪ {a}. Corollary 2.8. The sentence φ2 is א0-categorical, and hence is a complete sentence. Proof. Let M be a countable model. Enumerate I(M) as {ai | i < ω}. As we pointed out in Remark 2.4, a solution exists over the set {a0, a1} (any elements in the stalks Ga0,a1 and H ∗ a0,a1 work). By the extension property for solutions over finite sets we get a solution defined over the entire I(M). Hence φ2 is countably categorical by Lemma 2.6. We see that extension for solutions over finite sets translates into existence of solutions over countable sets. This is part of a general phenomenon that we describe below. We return to the general case k ≥ 2. Definition 2.9. Let M with M L′ = M(I) be a model of φ̂k. Let A be a subset of I(M) of size λ, and consider an arbitrary n-element set {b0, . . . , bn−1} ⊂ I . Suppose that, for each (n − 1)-element subset w of n = {0, . . . , n − 1}, we have a solution (gw, hw) over A ∪ {bl | l ∈ w} such that the solutions are compatible (i.e., ( ⋃ w gw, ⋃ w hw) is a function). We say that M has n-amalgamation for solutions over sets of size λ if for every such set A, there is a solution (g, h) over A ∪ {b0, . . . , bn−1} that simultaneously extends all the given solutions {(gw, hw) | w ∈ [n]n−1}. For n = 0 the given system of solutions is empty, thus 0-amalgamation over sets of size λ is existence for solutions over sets of size λ. For n = 1, the initial system of solutions degenerates to just (g∅, h∅), a solution on A; so the 1amalgamation property corresponds to the extension property for solutions. Generally, the number n in the statement of n-amalgamation property for solutions refers to the “dimension” of the system of solutions that we are able to amalgamate. Remark 2.10. Immediately from the definition we see that n-amalgamation for solutions of certain size implies m-amalgamation for solutions of the same size for any m < n. Indeed, we can obtain m-amalgamation by putting n−m elements of the set {b0, . . . , bn−1} inside A. Using Remark 2.4, we see that 2-amalgamation for solutions of size λ implies extension, and existence, of solutions of the same size. 10 JOHN T. BALDWIN AND ALEXEI KOLESNIKOV Lemma 2.11. The models φ̂k have the (k−1)-amalgamation property for solutions over finite sets. Proof. Enumerate A = {a0, . . . , ar−1}. We are given that ( ⋃ w gw, ⋃ w hw) is a function (where the union is over all w ∈ [k − 1]k−2). Moreover, it is a solution over W = ⋃ w dom(gw), (dom gw = [A ∪ {bi : i ∈ w}]k), since if u1, . . . uk+1 is a compatible (k + 1)-tuple of k-tuples from W , then each ui is in dom(gw) = dom(hw) for at least one w ∈ [k − 1]k−2. Denote the function ⋃ w gw by g−1. It is clear that in order to extend to a solution on A ∪ {b0, . . . , bk−2}, we only need to define the values (g, h) on the stalks {ai, b0, . . . , bk−2} for all i < r. For each i < r, let h(ai, b0, . . . , bk−2) be an arbitrary element of H∗ ai,b0,...,bk−2 . We need to check that (g−1, h) is still a solution. Remark 2.12. Hart and Shelah assert that categoricity holds up to אk−1; we show in Theorem 7.1 that this statement is incorrect. The Hart–Shelah argument breaks down at this very point. Their formulation of the analog of Lemma 2.11 asserts essentially the k, not k − 1, amalgamation property for solutions over finite sets. However, they did not make the compatibility requirement in Definition 2.9; and did not check that the function obtained after defining h is a partial solution. In fact, in their setting without the compatibility condition it need not be a solution, and there may not be a way of defining h to make (g−1, h) a solution. We present an example of the failure of 2-amalgamation for solutions over finite sets for models of φ2 at the end of this proof. As we will see in Lemma 2.14, (k − 1)-amalgamation for solutions over finite sets translates into existence of solutions, and hence categoricity, in אk−2. This is the reason for subscript of the categoricity cardinal being off by one in [9]. It is clear that (g−1, h) is a function with values in the appropriate stalks. To check that it is a solution, we need to make sure that we have not introduced new values that violate the predicate Q. This is easy: for each ai ∈ A, any compatible k + 1 tuple containing the k element set {ai, b0, . . . , bk−2} has to contain a k element set of the form {aj , b0, . . . , bk−2} for some j 6= i. Since the value g−1 at {aj , b0, . . . , bk−2} is not defined, there are simply no new compatible k + 1 tuples to worry about. Finally, we need to define g on the stalks of the form {ai, b0, . . . , bk−2}. We do it by induction on i < n, building an increasing chain of functions gi, i < n, with g0 extending g−1. Let {ws | s < k − 1} be an enumeration of all the k − 2 element subsets of k − 1; let bws denote the sequence {bi | i ∈ ws} and let cs,j = 〈a0, aj , bws〉. CATEGORICITY, AMALGAMATION, AND TAMENESS 11 For i = 0, pick an arbitrary starting point x ∈ Ga0,b0,...,bk−2 . Let γ0 ∈ G be such that for j = 1, . . . , n− 1 γ0(aj , b0, . . . , bk−2) = 1 if and only if M |= ¬Q(({a0, b0, . . . , bk−2}, x), g−1(c0,j), . . . , g−1(ck−1,j), h(aj , b0, . . . , bk−2)). Now we can extend the function g−1 to the function g0 by letting g0(a0, b0, . . . , bk−2) := ({a0, b0, . . . , bk−2}, x + γ0). It is clear that (g0, h) is a solution from its definition. For arbitrary i, suppose that the solution (gi−1, h) has been defined so that dom(gi−1) = dom(g−1) ∪ [{a0, . . . , ai−1, b0, . . . , bk−2}]. We need to extend gi−1 to a function gi, with domain dom(g−1) ∪ [{a0, . . . , ai, b0, . . . , bk−2}], by defining gi(ai, b0 . . . , bk−2). The strategy will be the same as before: we pick an arbitrary starting point and work to resolve all possible conflicts with the predicate Q. Let ds,j denote 〈ai, aj , bws〉. Pick an arbitrary starting point x ∈ Gai,b0,...,bk−2 . Let γi ∈ G be such that for j ∈ {0, . . . , n− 1} \ {i} γi(aj , b0, . . . , bk−2) = 1 if and only if M |= ¬Q(({ai, b0, . . . , bk−2}, x), g−1(d0,j), . . . , g−1(dk−1,j), h(aj , b0, . . . , bk−2)) and γi(u) = 0 if u ∈ dom(g−1)∪[{a0, . . . , ai, b0, . . . , bk−2}] is not of this form. For each (k− 2)-element set w of k− 1, let γw i ∈ G be such that for j < i γ i (ai, aj , bw) = 1 if and only if M |= ¬Q(({ai, b0, . . . , bk−2}, x), gi−1(aj , b0, . . . , bk−2), .., g−1(ds,j), .., h(ai, aj , bw)), and γ i (u) = 0 if u ∈ dom(g−1)∪[{a0, . . . , ai, b0, . . . , bk−2}] is not of this form, where ds,j ranges over all sequences 〈ai, aj , bws〉with ws a (k−2)-element subset of k − 1 except ws = w. The role of γw i is to avoid the conflict with the values already defined by gi−1. Notice that we have finitely many conditions to meet, so γi as well as γw i are all finite support functions in G.