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Non-forking formulas in Distal NIP theories Charlotte Kestner joint work with Gareth Boxall Department of Mathematics Imperial College London South Kensington London, UK c.kestner@imperial.ac.uk September 2, 2018 Charlotte Kestner joint

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Non-forking formulas in Distal NIP theories

Charlotte Kestner joint work with Gareth Boxall

Department of Mathematics Imperial College London South Kensington London, UK c.kestner@imperial.ac.uk

September 2, 2018

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Classifying Structures

Question What properties do definable sets have in a particular structure? Structure: (R, +, ×, 0, 1)

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Classifying Structures

Question What properties do definable sets have in a particular structure? Structure: (R, +, ×, 0, 1) Definable sets e.g. x < y defined by ∃z(x + z2 = y)

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Classifying Structures

Question What properties do definable sets have in a particular structure? Structure: (R, +, ×, 0, 1) Definable sets e.g. x < y defined by ∃z(x + z2 = y) O-minimal B.C. of intervals and points.

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Classifying Structures

Question What properties do definable sets have in a particular structure? Structure: (R, +, ×, 0, 1) Definable sets e.g. x < y defined by ∃z(x + z2 = y) O-minimal B.C. of intervals and points. Structure: (C, +, ×, 0, 1)

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Classifying Structures

Question What properties do definable sets have in a particular structure? Structure: (R, +, ×, 0, 1) Definable sets e.g. x < y defined by ∃z(x + z2 = y) O-minimal B.C. of intervals and points. Structure: (C, +, ×, 0, 1) Definable sets: B.C. of polynomial equalities.

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Classifying Structures

Question What properties do definable sets have in a particular structure? Structure: (R, +, ×, 0, 1) Definable sets e.g. x < y defined by ∃z(x + z2 = y) O-minimal B.C. of intervals and points. Structure: (C, +, ×, 0, 1) Definable sets: B.C. of polynomial equalities. strongly minimal definable sets in one variable finite

r cofinite

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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The Universe

O-min ·(R, +, ×, 0, 1) ·(R, +, ×, 0, 1, exp) Strongly Minimal ·(C, +, ×, 0, 1) · Vector spaces

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Stable theories

O-min ·(R, +, ×, 0, 1) ·(R, +, ×, 0, 1, exp) Strongly Minimal ·(C, +, ×, 0, 1) ·Vector spaces · Modules · Sep. closed fields Stable Definition A formula φ(x, y) is stable if there do not exist (ai)i∈ω, (bi)i∈ω such that M | = φ(ai, bj) if and only if i < j

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Indiscernible Sequences

Definition Given a tuple b and a set A the type of b over A is tp(b/A) = {φ(x, a) : M | = φ(b, a), a ∈ A}

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Indiscernible Sequences

Definition Given a tuple b and a set A the type of b over A is tp(b/A) = {φ(x, a) : M | = φ(b, a), a ∈ A} Definition A sequence (bi)i∈I is indiscernible over A if for every i0 < i1 < ... < in and j0 < j1 < ... < jn we have: tp(bi1....bin/A) = tp(bj1....bjn/A)

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Indiscernible Sequences

Definition Given a tuple b and a set A the type of b over A is tp(b/A) = {φ(x, a) : M | = φ(b, a), a ∈ A} Definition A sequence (bi)i∈I is indiscernible over A if for every i0 < i1 < ... < in and j0 < j1 < ... < jn we have: tp(bi1....bin/A) = tp(bj1....bjn/A) Example In (Q, <), see blackboard.

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Dividing formulas

Definition A formula φ(x, b) divides over A if there is an indiscernible sequence (bi)i∈ω with bo = b such that {φ(x, bi) : i ∈ ω} is inconsitent.

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Dividing formulas

Definition A formula φ(x, b) divides over A if there is an indiscernible sequence (bi)i∈ω with bo = b such that {φ(x, bi) : i ∈ ω} is inconsitent. Example

1 x = b 2 x = b Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Dividing formulas

Definition A formula φ(x, b) divides over A if there is an indiscernible sequence (bi)i∈ω with bo = b such that {φ(x, bi) : i ∈ ω} is inconsitent. Example

1 x = b 2 x = b

Remark In (C, +, ×, 0, 1) diving formulas give rise to a notion of rank that corresponds to transcendence degree.

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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The Universe

O-min ·(R, +, ×, 0, 1) ·(R, +, ×, 0, 1, exp) Strongly Minimal ·(C, +, ×, 0, 1) ·Vector spaces · Modules · Sep. closed fields Stable NIP · ACVF Definition A formula φ(x, y) is NIP if there is no infinite set A of |x|-tuples such that: (IP)φ,A for all A0 ⊆ A ∃bA0 such that φ(A, bA0) = A0.

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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NIP theories

Fact Let T be an NIP theory, M | = T, φ(x, y) formula then if C = {φ(x, b) : b ∈ M|y|}, the set system (M|x|, C) has finite (Vapnik – Chervonenkis) VC-dimension.

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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NIP theories

Fact Let T be an NIP theory, M | = T, φ(x, y) formula then if C = {φ(x, b) : b ∈ M|y|}, the set system (M|x|, C) has finite (Vapnik – Chervonenkis) VC-dimension. Theorem ( The (p, q)-theorem (Alon-Kleitman, Matousek)) Let p ≥ q be integers. Then there is an N ∈ Z such that the following holds: Let (X, S) be set system where every S ∈ S is non-empty. Assume: VC ∗(S) < q; For every p sets of S, some q have non-empty intersection. Then there is a subset of X of size N which intersects every element of S.

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Definable (p, q)

Question What nice behaviour of stable theories carries through to NIP?

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Definable (p, q)

Question What nice behaviour of stable theories carries through to NIP? Corollary (Chernikov-Simon) T NIP. Suppose φ(x, b) does not divide over M then we can find a formula ψ(y) ∈ tp(b/M) and a finite partition {Wj}n

j=1 of the set

ψ(M) such that {φ(x, bi) : bi ∈ Wi} is consistent. Conjecture (Definable (p, q) - conjecture) T NIP. The finite partition is not needed, i.e. Suppose φ(x, b) ∈ tp(b/M) does not divide over M then we can find a formula ψ(y) such that {φ(x, b) : b | = ψ(y)} is consistent.

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Definable (p, q)

Question What nice behaviour of stable theories carries through to NIP? Corollary (Chernikov-Simon) T NIP. Suppose φ(x, b) does not divide over M then we can find a formula ψ(y) ∈ tp(b/M) and a finite partition {Wj}n

j=1 of the set

ψ(M) such that {φ(x, bi) : bi ∈ Wi} is consistent. Conjecture (Definable (p, q) - conjecture) T NIP. The finite partition is not needed, i.e. Suppose φ(x, b) ∈ tp(b/M) does not divide over M then we can find a formula ψ(y) such that {φ(x, b) : b | = ψ(y)} is consistent. Fact Definable (p, q) - conjecture holds in Stable theories.

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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The Universe

O-min ·(R, +, ×, 0, 1) ·(R, +, ×, 0, 1, exp) Strongly Minimal ·(C, +, ×, 0, 1) ·Vector spaces · Modules · Sep. closed fields Stable NIP · ACVF Distal · Qp · Transerries Definition A theory T is distal if, for any small indiscernible sequence of the form I + {b} + J in M, and any small A ⊆ M, if I + J is indiscernible over A then I + {b} + J is indiscernible over A.

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Distal Theories

Example (Q, <) is Distal: See blackboard.

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Distal Theories

Example (Q, <) is Distal: See blackboard. Theorem (Chernikov-Starchenko) Graphs definable in a distal structure have the strong Erd¨

s-Hajnal

property. Definition Suppose R ⊆ Mm × Mn.

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Distal Theories

Example (Q, <) is Distal: See blackboard. Theorem (Chernikov-Starchenko) Graphs definable in a distal structure have the strong Erd¨

s-Hajnal

property. Definition Suppose R ⊆ Mm × Mn. A pair of subsets A ⊆ Mm, B ⊆ Mn is called R-homogeneous if either A × B ⊂ R or A × B ∩ R = ∅.

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Distal Theories

Example (Q, <) is Distal: See blackboard. Theorem (Chernikov-Starchenko) Graphs definable in a distal structure have the strong Erd¨

s-Hajnal

property. Definition Suppose R ⊆ Mm × Mn. A pair of subsets A ⊆ Mm, B ⊆ Mn is called R-homogeneous if either A × B ⊂ R or A × B ∩ R = ∅. We say a relation R satisifies the strong Erd¨

s -Hajnal property if

there is a constant δ = δ(R) > 0 such that for any finite subsets A, B there are A0 ⊂ A and B0 ⊂ B with |A0| > δ|A| |B0| > δ|B| such that (A0, B0) is R homogeneous.

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Theorem (Boxall-K) Definable (p, q) - conjecture holds in Distal theories.

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Theorem (Boxall-K) Definable (p, q) - conjecture holds in Distal theories. Question Does the Definable (p, q) - conjecture holds in NIP theories?

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Distal Expansions

Remark Distal theories are not closed under reducts (e.g. (M, =) is not distal).

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Distal Expansions

Remark Distal theories are not closed under reducts (e.g. (M, =) is not distal). Definition Let M | = T we define the Shelah expansion of M, MSh to be the expansion to the language LSh(M) containing for each externally definable (i.e. definable in an elementary extension of M) D ⊂ Mk a k-ary predicate RD(x).

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Distal Expansions

Remark Distal theories are not closed under reducts (e.g. (M, =) is not distal). Definition Let M | = T we define the Shelah expansion of M, MSh to be the expansion to the language LSh(M) containing for each externally definable (i.e. definable in an elementary extension of M) D ⊂ Mk a k-ary predicate RD(x). Theorem (Boxall-K) If MSh is distal if and only if M is distal (and thus any inbetween expansions is distal).

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories

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Thank you

Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories