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
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
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 + z 2 = y ) Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories
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 + z 2 = y ) O-minimal B.C. of intervals and points. Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories
Classifying Structures Question What properties do definable sets have in a particular structure? Structure: ( C , + , × , 0 , 1) Structure: ( R , + , × , 0 , 1) Definable sets e.g. x < y defined by ∃ z ( x + z 2 = y ) O-minimal B.C. of intervals and points. Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories
Classifying Structures Question What properties do definable sets have in a particular structure? Structure: ( C , + , × , 0 , 1) Structure: ( R , + , × , 0 , 1) Definable sets: B.C. of Definable sets e.g. x < y polynomial equalities. defined by ∃ z ( x + z 2 = y ) O-minimal B.C. of intervals and points. Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories
Classifying Structures Question What properties do definable sets have in a particular structure? Structure: ( C , + , × , 0 , 1) Structure: ( R , + , × , 0 , 1) Definable sets: B.C. of Definable sets e.g. x < y polynomial equalities. defined by ∃ z ( x + z 2 = y ) strongly minimal definable O-minimal B.C. of sets in one variable finite intervals and points. or cofinite Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories
The Universe O-min Strongly Minimal · ( R , + , × , 0 , 1 , exp) · Vector spaces · ( R , + , × , 0 , 1) · ( C , + , × , 0 , 1) Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories
Stable theories Stable · Modules · Sep. closed fields O-min Strongly Minimal · ( R , + , × , 0 , 1 , exp) · Vector spaces · ( R , + , × , 0 , 1) · ( C , + , × , 0 , 1) Definition A formula φ ( x , y ) is stable if there do not exist ( a i ) i ∈ ω , ( b i ) i ∈ ω such that M | = φ ( a i , b j ) if and only if i < j Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories
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
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 ( b i ) i ∈ I is indiscernible over A if for every i 0 < i 1 < ... < i n and j 0 < j 1 < ... < j n we have: tp ( b i 1 .... b i n / A ) = tp ( b j 1 .... b j n / A ) Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories
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 ( b i ) i ∈ I is indiscernible over A if for every i 0 < i 1 < ... < i n and j 0 < j 1 < ... < j n we have: tp ( b i 1 .... b i n / A ) = tp ( b j 1 .... b j n / A ) Example In ( Q , < ), see blackboard. Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories
Dividing formulas Definition A formula φ ( x , b ) divides over A if there is an indiscernible sequence ( b i ) i ∈ ω with b o = b such that { φ ( x , b i ) : i ∈ ω } is inconsitent. Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories
Dividing formulas Definition A formula φ ( x , b ) divides over A if there is an indiscernible sequence ( b i ) i ∈ ω with b o = b such that { φ ( x , b i ) : 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
Dividing formulas Definition A formula φ ( x , b ) divides over A if there is an indiscernible sequence ( b i ) i ∈ ω with b o = b such that { φ ( x , b i ) : 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
The Universe NIP · ACVF Stable · Modules · Sep. closed fields O-min Strongly Minimal · ( R , + , × , 0 , 1 , exp) · Vector spaces · ( R , + , × , 0 , 1) · ( C , + , × , 0 , 1) 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 A 0 ⊆ A ∃ b A 0 such that φ ( A , b A 0 ) = A 0 . Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories
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
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
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
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 { W j } n j =1 of the set ψ ( M ) such that { φ ( x , b i ) : b i ∈ W i } 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
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 { W j } n j =1 of the set ψ ( M ) such that { φ ( x , b i ) : b i ∈ W i } 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
The Universe NIP · ACVF Distal Stable · Q p · Modules · Transerries · Sep. closed fields O-min Strongly Minimal · ( R , + , × , 0 , 1 , exp) · Vector spaces · ( R , + , × , 0 , 1) · ( C , + , × , 0 , 1) 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
Distal Theories Example ( Q , < ) is Distal: See blackboard. Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories
Distal Theories Example ( Q , < ) is Distal: See blackboard. Theorem (Chernikov-Starchenko) Graphs definable in a distal structure have the strong Erd¨ os-Hajnal property. Definition Suppose R ⊆ M m × M n . Charlotte Kestner joint work with Gareth Boxall Non-forking formulas in Distal NIP theories
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