Omega-categorical structures and Macpherson-Steinhorn measurability. - - PowerPoint PPT Presentation

Omega-categorical structures and Macpherson-Steinhorn measurability.

David Evans

• Dept. of Mathematics, Imperial College London.

FPS, Leeds, April 2018.

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• 1. MS-measurability.

For a (first-order) L-structure M we denote by Def(M) the collection of all non-empty parameter definable subsets of Mn (for all n ≥ 1). DEF: A structure M is MS-measurable if there is a dimension - measure function h : Def(M) → N × R>0 satisfying the following, where we write h(X) = (dim(X), ν(X)): (i) If X is finite (and non-empty) then h(X) = (0, |X|); (ii) For every formula φ(¯ x, ¯ y) there is a finite set Dφ ⊆ N × R>0 of possible vaules for h(φ(¯ x, ¯ a)) (with ¯ a ∈ Mn) and for each such value, the set of ¯ a giving this value is 0-definable; (iii) (Fubini property) Suppose X, Y ∈ Def(M) and f : X → Y is a definable surjection. By (ii), Y can be partitioned into disjoint definable sets Y1, . . . , Yr such that h(f −1(y)) is constant, equal to (di, mi), for y ∈ Yi. Let h(Yi) = (ei, ni). Let c be the maximum of di + ei and suppose this is attained for i = 1, . . . , s. Then h(X) = (c, m1n1 + · · · + msns).

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Measures

Suppose h = (dim, ν) is a dimension-measure function for M. OBSERVATIONS: MS-measurability is a property of Th(M). May assume ν(M) = 1, so h(Mn) = (n dim(M), 1). (R. Elwes, M. Ryten) M is supersimple of finite rank. M is (weak) unimodular:

◮ If fi : X → Y are ki-to-1 definable surjections (in Meq), then k1 = k2.

Let Bn(M) denote the definable subsets of Mn. If D ∈ Bn(M), let µn(D) = ν(D) if dim(D) = dim(Mn)

• therwise

. This is a finitely-additive, definable probability measure on Bn(M). If L is countable and M is ℵ1-saturated, then µn extends uniquely to a measure µn on Bσ

n(M), the σ-algebra generated by Bn(M).

Fubini’s Theorem holds for the µn.

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Measures

Suppose h = (dim, ν) is a dimension-measure function for M. OBSERVATIONS: MS-measurability is a property of Th(M). May assume ν(M) = 1, so h(Mn) = (n dim(M), 1). (R. Elwes, M. Ryten) M is supersimple of finite rank. M is (weak) unimodular:

◮ If fi : X → Y are ki-to-1 definable surjections (in Meq), then k1 = k2.

Let Bn(M) denote the definable subsets of Mn. If D ∈ Bn(M), let µn(D) = ν(D) if dim(D) = dim(Mn)

• therwise

. This is a finitely-additive, definable probability measure on Bn(M). If L is countable and M is ℵ1-saturated, then µn extends uniquely to a measure µn on Bσ

n(M), the σ-algebra generated by Bn(M).

Fubini’s Theorem holds for the µn.

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Measures

Suppose h = (dim, ν) is a dimension-measure function for M. OBSERVATIONS: MS-measurability is a property of Th(M). May assume ν(M) = 1, so h(Mn) = (n dim(M), 1). (R. Elwes, M. Ryten) M is supersimple of finite rank. M is (weak) unimodular:

◮ If fi : X → Y are ki-to-1 definable surjections (in Meq), then k1 = k2.

Let Bn(M) denote the definable subsets of Mn. If D ∈ Bn(M), let µn(D) = ν(D) if dim(D) = dim(Mn)

• therwise

. This is a finitely-additive, definable probability measure on Bn(M). If L is countable and M is ℵ1-saturated, then µn extends uniquely to a measure µn on Bσ

n(M), the σ-algebra generated by Bn(M).

Fubini’s Theorem holds for the µn.

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Examples

(Z. Chatzidakis, L. van den Dries, A. Macintyre) Pseudofinite fields. (H. D. Macpherson, C. Steinhorn) Ultraproducts of asymptotic classes of finite structures:

◮ Random graph ◮ Smoothly approximated structures ◮ .... 4 / 18

Examples

(Z. Chatzidakis, L. van den Dries, A. Macintyre) Pseudofinite fields. (H. D. Macpherson, C. Steinhorn) Ultraproducts of asymptotic classes of finite structures:

◮ Random graph ◮ Smoothly approximated structures ◮ .... 4 / 18

QUESTIONS: (R. Elwes, H. D. Macpherson)

• 1. Is an ω-categorical, MS-measurable structure necessarily
• ne-based?
• 2. Is there an example of a supersimple, finite rank, unimodular

structure which is not MS-measurable? REMARKS: ω-categoricity implies unimodularity. Examples of non-one-based, ω-categorical, supersimple structures: Hrushovski constructions (1988, 1997). QUESTION′: Are any of the ω-categorical Hrushovski constructions MS-measurable?

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QUESTIONS: (R. Elwes, H. D. Macpherson)

• 1. Is an ω-categorical, MS-measurable structure necessarily
• ne-based?
• 2. Is there an example of a supersimple, finite rank, unimodular

structure which is not MS-measurable? REMARKS: ω-categoricity implies unimodularity. Examples of non-one-based, ω-categorical, supersimple structures: Hrushovski constructions (1988, 1997). QUESTION′: Are any of the ω-categorical Hrushovski constructions MS-measurable?

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QUESTIONS: (R. Elwes, H. D. Macpherson)

• 1. Is an ω-categorical, MS-measurable structure necessarily
• ne-based?
• 2. Is there an example of a supersimple, finite rank, unimodular

structure which is not MS-measurable? REMARKS: ω-categoricity implies unimodularity. Examples of non-one-based, ω-categorical, supersimple structures: Hrushovski constructions (1988, 1997). QUESTION′: Are any of the ω-categorical Hrushovski constructions MS-measurable?

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QUESTIONS: (R. Elwes, H. D. Macpherson)

• 1. Is an ω-categorical, MS-measurable structure necessarily
• ne-based?
• 2. Is there an example of a supersimple, finite rank, unimodular

structure which is not MS-measurable? REMARKS: ω-categoricity implies unimodularity. Examples of non-one-based, ω-categorical, supersimple structures: Hrushovski constructions (1988, 1997). QUESTION′: Are any of the ω-categorical Hrushovski constructions MS-measurable?

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QUESTIONS: (R. Elwes, H. D. Macpherson)

• 1. Is an ω-categorical, MS-measurable structure necessarily
• ne-based?
• 2. Is there an example of a supersimple, finite rank, unimodular

structure which is not MS-measurable? REMARKS: ω-categoricity implies unimodularity. Examples of non-one-based, ω-categorical, supersimple structures: Hrushovski constructions (1988, 1997). QUESTION′: Are any of the ω-categorical Hrushovski constructions MS-measurable?

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A modest result

Theorem A

There is an ω-categorical supersimple structure of SU-rank 1 which is not MS-measurable. – The example is a Hrushovski construction. – Possibly the proof is over-elaborate.

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A modest result

Theorem A

There is an ω-categorical supersimple structure of SU-rank 1 which is not MS-measurable. – The example is a Hrushovski construction. – Possibly the proof is over-elaborate.

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• 2. A higher amalgamation result

NOTATION: L countable; M an L-structure. If k ≤ n ∈ N, denote by [n]k the k-subsets of [n] = {1, . . . , n}. If I ∈ [n]k, then πI : Mn → Mk denotes the projection to coordinates in I.

Theorem B

Suppose M is a MS-measurable structure and E ⊆ Mn is a definable

• subset. Suppose that:

(a) dim(πJ(E)) = dim(Mn−1), where J = {1, . . . , n − 1}, and (b) if (a1, . . . , an) ∈ E, then ai ∈ acl({aj : j = i}) (for all i ≤ n). Then dim({¯ b ∈ Mn : πI(¯ b) ∈ πI(E) for all I ∈ [n]n−1}) = dim(Mn). REMARKS: 1. Is there a relation between Theorem B and independent n-amalgamation over a model?

• 2. Do MS-measurable structures satisfy independent n-amalgamation
• ver a model?

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• 2. A higher amalgamation result

NOTATION: L countable; M an L-structure. If k ≤ n ∈ N, denote by [n]k the k-subsets of [n] = {1, . . . , n}. If I ∈ [n]k, then πI : Mn → Mk denotes the projection to coordinates in I.

Theorem B

Suppose M is a MS-measurable structure and E ⊆ Mn is a definable

• subset. Suppose that:

(a) dim(πJ(E)) = dim(Mn−1), where J = {1, . . . , n − 1}, and (b) if (a1, . . . , an) ∈ E, then ai ∈ acl({aj : j = i}) (for all i ≤ n). Then dim({¯ b ∈ Mn : πI(¯ b) ∈ πI(E) for all I ∈ [n]n−1}) = dim(Mn). REMARKS: 1. Is there a relation between Theorem B and independent n-amalgamation over a model?

• 2. Do MS-measurable structures satisfy independent n-amalgamation
• ver a model?

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• 2. A higher amalgamation result

NOTATION: L countable; M an L-structure. If k ≤ n ∈ N, denote by [n]k the k-subsets of [n] = {1, . . . , n}. If I ∈ [n]k, then πI : Mn → Mk denotes the projection to coordinates in I.

Theorem B

Suppose M is a MS-measurable structure and E ⊆ Mn is a definable

• subset. Suppose that:

(a) dim(πJ(E)) = dim(Mn−1), where J = {1, . . . , n − 1}, and (b) if (a1, . . . , an) ∈ E, then ai ∈ acl({aj : j = i}) (for all i ≤ n). Then dim({¯ b ∈ Mn : πI(¯ b) ∈ πI(E) for all I ∈ [n]n−1}) = dim(Mn). REMARKS: 1. Is there a relation between Theorem B and independent n-amalgamation over a model?

• 2. Do MS-measurable structures satisfy independent n-amalgamation
• ver a model?

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Deducing Thm A

1

Produce an ω-categorical Hrushovski structure Mf of SU-rank 1 which fails to have the amalgamation property in Theorem B for dim equal to SU-rank.

2

Show that if (dim, ν) is a dimension-measure function on Def(Mf) then dim is equal to SU-rank (after normalising).

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Deducing Thm A

1

Produce an ω-categorical Hrushovski structure Mf of SU-rank 1 which fails to have the amalgamation property in Theorem B for dim equal to SU-rank.

2

Show that if (dim, ν) is a dimension-measure function on Def(Mf) then dim is equal to SU-rank (after normalising).

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Overkill?

Theorem B is deduced from an infinitary version of the Hypergraph Removal Lemma (Gowers; Nagle, Rödl, Schacht; Skokan), due to Towsner and Goldbring + Towsner (following work of Tao). HRL implies Szemerédi’s Theorem and is a generalisation of: THEOREM: (Triangle Removal Lemma, Rusza+Szemerédi) For every ǫ > 0 there is δ > 0 with the property that: for every finite graph Γ = (V; E) with ≤ δ|V|3 triangles, there is E′ ⊆ E of size ≤ ǫ|V|2, such that Γ′ = (V; E \ E′) is triangle-free. SO: if removing ǫ|V|2 edges from Γ does not create a triangle-free graph, then Γ contains at least δ|V|3 triangles.

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Overkill?

Theorem B is deduced from an infinitary version of the Hypergraph Removal Lemma (Gowers; Nagle, Rödl, Schacht; Skokan), due to Towsner and Goldbring + Towsner (following work of Tao). HRL implies Szemerédi’s Theorem and is a generalisation of: THEOREM: (Triangle Removal Lemma, Rusza+Szemerédi) For every ǫ > 0 there is δ > 0 with the property that: for every finite graph Γ = (V; E) with ≤ δ|V|3 triangles, there is E′ ⊆ E of size ≤ ǫ|V|2, such that Γ′ = (V; E \ E′) is triangle-free. SO: if removing ǫ|V|2 edges from Γ does not create a triangle-free graph, then Γ contains at least δ|V|3 triangles.

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Overkill?

Theorem B is deduced from an infinitary version of the Hypergraph Removal Lemma (Gowers; Nagle, Rödl, Schacht; Skokan), due to Towsner and Goldbring + Towsner (following work of Tao). HRL implies Szemerédi’s Theorem and is a generalisation of: THEOREM: (Triangle Removal Lemma, Rusza+Szemerédi) For every ǫ > 0 there is δ > 0 with the property that: for every finite graph Γ = (V; E) with ≤ δ|V|3 triangles, there is E′ ⊆ E of size ≤ ǫ|V|2, such that Γ′ = (V; E \ E′) is triangle-free. SO: if removing ǫ|V|2 edges from Γ does not create a triangle-free graph, then Γ contains at least δ|V|3 triangles.

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• 3. Hypergraph Removal

SET-UP: (Vague, but ok for MS-measurable strs.) L countable; N an ℵ1-saturated L-structure M a countable elementary substructure For n ∈ N, µn is a definable probability measure on Bσ

n(M) and

these satisfy Fubini. NOTATION: If I ∈ [n]k, then Bn,I(M) consists of the M-definable subsets D of Nn where only coordinates in I are involved in the defining

• formula. So D = π−1

I

(E) for some E ∈ Bk(M).

Theorem (H. Towsner; I. Goldbring + H. Towsner)

Suppose k < n and for each I ∈ [n]k we have AI ∈ Bσ

n,I(M).

Suppose that there is ǫ > 0 such that whenever BI ∈ Bn,I(M) and µn(AI \ BI) < ǫ (for I ∈ [n]k), then

I∈[n]k BI = ∅.

Then µn(

I∈[n]k AI) > 0.

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A corollary

The following property follows from the Fubini property for MS-measurable structures: (CP) Suppose A, B ∈ Bm(N) and C ⊆ A × B is definable and projects

• nto each factor, with fibres of size at most k. Then

µm(B) ≤ kµm(A).

Corollary (to HRL)

Suppose (CP) holds. Let J = {1, . . . , n − 1} . Suppose E ∈ Bn(N) is such that µn−1(πJ(E)) > 0 and if (a1, . . . , an) ∈ E, then ai ∈ acl({aj : j = i}) (for all i ≤ n). Then µn({¯ b ∈ Nn : πI(¯ b) ∈ πI(E) for all I ∈ [n]n−1}) > 0. – Theorem B follows from this.

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A corollary

The following property follows from the Fubini property for MS-measurable structures: (CP) Suppose A, B ∈ Bm(N) and C ⊆ A × B is definable and projects

• nto each factor, with fibres of size at most k. Then

µm(B) ≤ kµm(A).

Corollary (to HRL)

Suppose (CP) holds. Let J = {1, . . . , n − 1} . Suppose E ∈ Bn(N) is such that µn−1(πJ(E)) > 0 and if (a1, . . . , an) ∈ E, then ai ∈ acl({aj : j = i}) (for all i ≤ n). Then µn({¯ b ∈ Nn : πI(¯ b) ∈ πI(E) for all I ∈ [n]n−1}) > 0. – Theorem B follows from this.

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A corollary

The following property follows from the Fubini property for MS-measurable structures: (CP) Suppose A, B ∈ Bm(N) and C ⊆ A × B is definable and projects

• nto each factor, with fibres of size at most k. Then

µm(B) ≤ kµm(A).

Corollary (to HRL)

Suppose (CP) holds. Let J = {1, . . . , n − 1} . Suppose E ∈ Bn(N) is such that µn−1(πJ(E)) > 0 and if (a1, . . . , an) ∈ E, then ai ∈ acl({aj : j = i}) (for all i ≤ n). Then µn({¯ b ∈ Nn : πI(¯ b) ∈ πI(E) for all I ∈ [n]n−1}) > 0. – Theorem B follows from this.

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• 4. Hrushovski’s ω-categorical construction I

Relational language L: {Ri : i ≤ s}, where R1 is 3-ary. C: class of finite L-strs If C ⊆ A ∈ C let δ(C) = |C| − Σi|Ri[C]|. (Predimension of C.) If A ⊆ B ∈ C write A ≤ B if δ(X) > δ(A) whenever A ⊂ X ⊆ B. LEMMA:

◮ If A ≤ B and X ⊆ B then A ∩ X ≤ X; ◮ If A ≤ B ≤ C then A ≤ C. ◮ If A1, A2 ≤ B then A1 ∩ A2 ≤ B.

d-closure clB

d (X) = {A : X ⊆ A ≤ B} ≤ B.

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• 4. Hrushovski’s ω-categorical construction I

Relational language L: {Ri : i ≤ s}, where R1 is 3-ary. C: class of finite L-strs If C ⊆ A ∈ C let δ(C) = |C| − Σi|Ri[C]|. (Predimension of C.) If A ⊆ B ∈ C write A ≤ B if δ(X) > δ(A) whenever A ⊂ X ⊆ B. LEMMA:

◮ If A ≤ B and X ⊆ B then A ∩ X ≤ X; ◮ If A ≤ B ≤ C then A ≤ C. ◮ If A1, A2 ≤ B then A1 ∩ A2 ≤ B.

d-closure clB

d (X) = {A : X ⊆ A ≤ B} ≤ B.

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• 4. Hrushovski’s ω-categorical construction I

Relational language L: {Ri : i ≤ s}, where R1 is 3-ary. C: class of finite L-strs If C ⊆ A ∈ C let δ(C) = |C| − Σi|Ri[C]|. (Predimension of C.) If A ⊆ B ∈ C write A ≤ B if δ(X) > δ(A) whenever A ⊂ X ⊆ B. LEMMA:

◮ If A ≤ B and X ⊆ B then A ∩ X ≤ X; ◮ If A ≤ B ≤ C then A ≤ C. ◮ If A1, A2 ≤ B then A1 ∩ A2 ≤ B.

d-closure clB

d (X) = {A : X ⊆ A ≤ B} ≤ B.

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• 4. Hrushovski’s ω-categorical construction I

Relational language L: {Ri : i ≤ s}, where R1 is 3-ary. C: class of finite L-strs If C ⊆ A ∈ C let δ(C) = |C| − Σi|Ri[C]|. (Predimension of C.) If A ⊆ B ∈ C write A ≤ B if δ(X) > δ(A) whenever A ⊂ X ⊆ B. LEMMA:

◮ If A ≤ B and X ⊆ B then A ∩ X ≤ X; ◮ If A ≤ B ≤ C then A ≤ C. ◮ If A1, A2 ≤ B then A1 ∩ A2 ≤ B.

d-closure clB

d (X) = {A : X ⊆ A ≤ B} ≤ B.

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Hrushovski’s construction II

f : R≥0 → R≥0 an increasing (differentiable) function which tends to infinity. Let Cf = {A ∈ C : δ(Y) ≥ f(|Y|) for all Y ⊆ A}. If X ⊆ B ∈ Cf, then |clB

d (X)| ≤ f −1(δ(X)) ≤ f −1(|X|).

For suitable f the class (Cf, ≤) has free amalgamation over ≤-substructures. E.g. f ′(x) ≤ 1/(1 + x) and non-increasing. Assume this henceforth. In this case the Fraïssé limit construction gives a countable structure Mf characterised by:

◮ Mf is the union of a chain of finite ≤-substrs; ◮ every structure in Cf is isomorphic to a ≤-substr of Mf; ◮ isomorphisms between finite ≤-substrs of Mf extend to

automorphisms.

The structure Mf is ω-categorical and acl is given by cld.

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Hrushovski’s construction II

f : R≥0 → R≥0 an increasing (differentiable) function which tends to infinity. Let Cf = {A ∈ C : δ(Y) ≥ f(|Y|) for all Y ⊆ A}. If X ⊆ B ∈ Cf, then |clB

d (X)| ≤ f −1(δ(X)) ≤ f −1(|X|).

For suitable f the class (Cf, ≤) has free amalgamation over ≤-substructures. E.g. f ′(x) ≤ 1/(1 + x) and non-increasing. Assume this henceforth. In this case the Fraïssé limit construction gives a countable structure Mf characterised by:

◮ Mf is the union of a chain of finite ≤-substrs; ◮ every structure in Cf is isomorphic to a ≤-substr of Mf; ◮ isomorphisms between finite ≤-substrs of Mf extend to

automorphisms.

The structure Mf is ω-categorical and acl is given by cld.

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Dimension and supersimplicity

DEF: For X ⊆ Mf, the dimension of X is d(X) = δ(cld(X)). Also let d(Y/X) = d(Y ∪ X) − d(X). THEOREM: (E. Hrushovski) If additionally f(3x) ≤ f(x) + 1, then Mf is supersimple and for all tuples ¯ a, ¯ b in Mf, SU(tp(¯ a/¯ b)) = d(¯ a/¯ b).

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dim(.) = d(.)

Proposition

Suppose f ′(x) ≤ 1/2(x + 1) and Mf has a dimension - measure function (dim, ν).

1

Let ¯ a, b ∈ Mf with b ∈ acl(¯ a) and P the realisations of tp(b/¯ a). Let r ∈ N. Then for every ¯ y ∈ Mr

f , there is some ¯

x ∈ Pr+2 with ¯ y ∈ acl(¯ x¯ a).

2

If ¯ a, ¯ b are tuples in Mf, then dim(¯ b/¯ a) = dim(Mf)d(¯ b/¯ a). PROOF OF (1) ⇒ (2): Enough to prove (2) when ¯ b = b ∈ Mf \ acl(¯ a). So we must show dim(P) = dim(Mf). We have dim

• {acl(¯

x¯ a)r : ¯ x ∈ P2+r} ≤ (2 + r)dim(P), (where acl(¯ x¯ a)r is the set of r-tuples from acl(¯ x¯ a)). So by (1): rdim(Mf) = dim(Mr

f ) ≤ (2 + r)dim(P) ≤ (2 + r)dim(Mf).

Dividing by (2 + r) and letting r → ∞ gives dim(P) = dim(Mf). ✷

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Aside: pseudofiniteness?

It is a well-known open problem to determine the pseudofiniteness or

• therwise of the ω-categorical Hrushovski constructions.

Corollary (to (1) in Proposition)

Suppose Mf is elementarily equivalent to an ultraproduct of finite structures (Fi : i < ω). Suppose ¯ b ∈ Mn

f and ¯

a is a tuple in Mf. Let Φ(¯ x) ∈ tp(¯ b/¯ a) isolate tp(¯ b/¯ a). Then the following are equivalent:

1

d(¯ b/¯ a) < n;

2

There exists some ǫ > 0 such that for almost all i: log |Φ(Fi)| ≤ (1 − ǫ) log |F n

i |.

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The example I

WANT: Mf and a (non-empty) 0-definable E ⊆ Mn

f such that:

(1) if (a1, . . . , an) ∈ E, then d(a1, . . . , an) = n − 1, and (2) there is no ¯ b ∈ Mn

f with d(¯

b) = n such that for all I ∈ [n]n−1 we have πI(¯ b) ∈ πI(E). TAKE: L has relations: R, 3-ary; S, 10-ary; U, 11-ary. f(x) = log8(x + 1).

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The example II

E ⊆ M10

f

consists of tuples ¯ a = (a1, . . . , a10) such that: S(a1, . . . , a10) holds (and no other relations) There are distinct ui with U(a1, . . . , a10, ui) for i ≤ r = 89 − 11. cld(¯ a) = ¯ a ∪ {ui : i ≤ r}. NOTE: As δ(¯ a) = δ(cld(¯ a)) = 9 and |cld(¯ a)| = 89 − 1, it follows that E = ∅. Suppose ¯ b = (b1, . . . , b10) is such that πI(¯ b) ∈ πI(E) for all I ∈ [10]9 and d(¯ b) = 10. Let B = cld(¯ b). So ¯ bI | ⌣ ¯

bI∩J

¯ bJ for I, J ∈ [10]9 and therefore cld(¯ bI) ∩ cld(¯ bJ) = ¯ bI∩J if I = J. Thus: |B| ≥ 10 + 10(1 + 89 − 11) so f(|B|) ≥ log8(10.89 − 89) > 10 = δ(B). Contradiction.

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The example II

E ⊆ M10

f

consists of tuples ¯ a = (a1, . . . , a10) such that: S(a1, . . . , a10) holds (and no other relations) There are distinct ui with U(a1, . . . , a10, ui) for i ≤ r = 89 − 11. cld(¯ a) = ¯ a ∪ {ui : i ≤ r}. NOTE: As δ(¯ a) = δ(cld(¯ a)) = 9 and |cld(¯ a)| = 89 − 1, it follows that E = ∅. Suppose ¯ b = (b1, . . . , b10) is such that πI(¯ b) ∈ πI(E) for all I ∈ [10]9 and d(¯ b) = 10. Let B = cld(¯ b). So ¯ bI | ⌣ ¯

bI∩J

¯ bJ for I, J ∈ [10]9 and therefore cld(¯ bI) ∩ cld(¯ bJ) = ¯ bI∩J if I = J. Thus: |B| ≥ 10 + 10(1 + 89 − 11) so f(|B|) ≥ log8(10.89 − 89) > 10 = δ(B). Contradiction.

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