Definable Regularity for NIP Relations Roland Walker University of Illinois at Chicago April 19, 2017 Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 1 / 38
(Simple) Graphs Let V be a finite set, and let E ⊆ V × V . Definition If E is irreflexive and symmetric, we call G = ( V , E ) a (simple) graph with • vertex set V ( G ) := V • order v ( G ) := | V | • edge set E ( G ) := {{ a , b } : ( a , b ) ∈ E } • size e ( G ) := | E ( G ) | = | E | / 2 Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 2 / 38
Bipartite Graphs Let V and W be finite sets, and let E ⊆ V × W . Definition We say that G = ( V , W ; E ) is a bipartite graph with • vertex set V ( G ) := V + W • order v ( G ) := | V | + | W | • edge set E ( G ) := E • size e ( G ) := | E | Note: Any graph G = ( V , E ) induces a bipartite graph G ′ = ( V , V ; E ). Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 3 / 38
Edge Density for Induced Bipartite Graphs Let G = ( V , E ) be a finite graph. Definition Given A , B ⊆ V , the induced bipartite graph ( A , B ; E ) has • vertex set V ( A , B ) := A + B • order v ( A , B ) := | A | + | B | • edge set E ( A , B ) := E ∩ ( A × B ) • size e ( A , B ) := | E ( A , B ) | d ( A , B ) := e ( A , B ) • density | A || B | Note: We do not require A and B to be disjoint. Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 4 / 38
Regularity and Defect Let G = ( V , E ) be a finite graph. Fix ε, δ ∈ [0 , 1]. Definition Given A , B ⊆ V , we say the pair ( A , B ) is ( ε, δ ) -regular iff: there exists α ∈ [0 , 1] such that for all nonempty sets A ′ ⊆ A and B ′ ⊆ B with | A ′ | ≥ δ | A | and | B ′ | ≥ δ | B | , we have | d ( A ′ , B ′ ) − α | ≤ ε 2 . Let P be a finite partition of V . Fix η ∈ [0 , 1]. Definition The defect of P is def ε,δ ( P ) := { ( A , B ) ∈ P 2 : ( A , B ) not ( ε, δ )-regular } , and we say that P is ( ε, δ, η )-regular iff: � | A || B | ≤ η | V | 2 . ( A , B ) ∈ def ε,δ ( P ) Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 5 / 38
Szemer´ edi Regularity Lemma (without Equipartition) Lemma For all ε, δ, η > 0 , there exists M = M ( ε, δ, η ) such that any finite graph has an ( ε, δ, η ) -regular partition with at most M parts. Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 6 / 38
Szemer´ edi Regularity Lemma (without Equipartition) Lemma For all ε, δ, η > 0 , there exists M = M ( ε, δ, η ) such that any finite graph has an ( ε, δ, η ) -regular partition with at most M parts. edi 1976) M ( ε, ε, ε ) ≤ twr 2 ( O ( ε − 5 )) (Szemer´ Can irregular pairs be completely eliminated? No, if we admit arbitrarily large half-graphs, then irregular pairs are necessary. Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 6 / 38
Szemer´ edi Regularity Lemma (without Equipartition) Lemma For all ε, δ, η > 0 , there exists M = M ( ε, δ, η ) such that any finite graph has an ( ε, δ, η ) -regular partition with at most M parts. edi 1976) M ( ε, ε, ε ) ≤ twr 2 ( O ( ε − 5 )) (Szemer´ Can irregular pairs be completely eliminated? No, if we admit arbitrarily large half-graphs, then irregular pairs are necessary. How fast does M grow as δ → 0? (Gowers 1997) M (1 − δ 1 / 16 , δ, 1 − 20 δ 1 / 16 ) ≥ twr 2 (Ω( δ − 1 / 16 )) How fast does M grow as η → 0? (Conlon-Fox 2012) ∃ ε, δ > 0 such that M ( ε, δ, η ) ≥ twr 2 (Ω( η − 1 )) Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 6 / 38
Bipartite Regularity and Defect Let G = ( V 1 , V 2 ; E ) be a finite bipartite graph. Let P = ( P 1 , P 2 ) where each P i is a finite partition of V i . We will use the � P � to denote max( | P 1 | , | P 2 | ). Let ε, δ, η ∈ [0 , 1] . Definition The defect of P is def ε,δ ( P ) := { ( A , B ) ∈ P 1 × P 2 : ( A , B ) not ( ε, δ )-regular } , and we call P ( ε, δ, η )-regular iff: � | A || B | ≤ η | V 1 || V 2 | . ( A , B ) ∈ def ε,δ ( P ) Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 7 / 38
Bipartite Szemer´ edi Regularity Lemma Lemma For all ε, δ, η > 0 , there exists M = M ( ε, δ, η ) such that any finite bipartite graph has an ( ε, δ, η ) -regular partition P with � P � ≤ M. (Gowers 1997) ⇒ M (1 − δ 1 / 16 , δ, 8 9 − 40 δ 1 / 16 ) ≥ twr 2 (Ω( δ − 1 / 16 )) Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 8 / 38
k -Partite k -Uniform Hypergraphs Let k ≥ 2, V 1 , . . . , V k be finite sets, and E ⊆ V 1 × · · · × V k . Definition We say that G = ( V 1 , · · · , V k ; E ) is a k-partite (k-uniform) hypergraph with • vertex set V ( G ) := V 1 + · · · + V k • order v ( G ) := | V 1 | + · · · + | V k | • edge set E ( G ) := E • size e ( G ) := | E | Note: When k = 2, G = ( V 1 , V 2 ; E ) is a bipartite graph. Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 9 / 38
Edge Density for k -Partite Hypergraphs Let G = ( V 1 , . . . , V k ; E ) be a finite k -partite hypergraph. Definition Given A i ⊆ V i , the subgraph ( A 1 , . . . , A k ; E ) has • vertex set V ( A 1 , . . . , A k ) := A 1 + · · · + A k • order v ( A 1 , . . . , A k ) := | A 1 | + · · · + | A k | • edge set E ( A 1 , . . . , A k ) := E ∩ ( A 1 × · · · × A k ) • size e ( A 1 , . . . , A k ) := | E ( A 1 , . . . , A k ) | d ( A 1 , . . . , A k ) := e ( A 1 , . . . , A k ) • density | A 1 | · · · | A k | Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 10 / 38
Rectangular Sets Definition A tuple of sets A = ( A 1 , . . . , A k ) names the rectangular set A 1 × · · · × A k . We write B ⊆ A iff: B ⊆ A 1 × · · · × A k . We write B ⊆ A iff: each B i ⊆ A i . We use | A | to denote | A 1 | · · · | A k | . We use � A � to denote max( | A i | : 1 ≤ i ≤ k ). Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 11 / 38
k -Partite Regularity and Defect Let G = ( V 1 , . . . , V k ; E ) be a finite k -partite hypergraph. Fix ε, δ, η ∈ [0 , 1]. Definition Given A ⊆ V , we say A is ( ε, δ ) -regular iff: there exists α ∈ [0 , 1] such that for all nonempty B ⊆ A with | B i | ≥ δ | A i | , we have | d ( B ) − α | ≤ ε 2 . Let P = ( P 1 , . . . , P k ) where each P i is a finite partition of V i . Definition The defect of P is def ε,δ ( P ) := { A ∈ P : A not ( ε, δ )-regular } , and we call P ( ε, δ, η )-regular iff: � | A | ≤ η | V | . ( A , B ) ∈ def ε,δ ( P ) Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 12 / 38
Fibers and VC Dimension Let E ⊆ V 1 × · · · × V k . For each I ⊆ [ k ], let V I denote � i ∈ I V i . With I ⊆ [ k ] specified, we can view E as a subset of V I × V I c and for each b ∈ V I c , let E b denote the fiber of b ; i.e., E b := { a ∈ V I : ( a , b ) ∈ E } . Definition VC( E ) = max { VC( S I ) : I ⊆ [ k ] } where S I = { E b : b ∈ V I c } . Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 13 / 38
Regularity Lemma for k -Partite Hypergraphs Fix k ≥ 2 and d ∈ N . Lemma For any ε, δ, η > 0 , there is a constant c = c ( k , d ) such that any finite k-partite hypergraph with VC dimension at most d has an ( ε, δ, η ) -regular partition P with � P � ≤ O ( γ − c ) where γ = min { ε, δ, η } . Note: When k = 2, this is the Bipartite Szemer´ edi Regularity Lemma restricted to graphs with VC( E ) ≤ d . Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 14 / 38
Finitely Additive Probability Measures Let X be a set, B ⊆ P ( X ) a boolean algebra, and µ : B → [0 , 1]. Definition We call µ a finitely additive probability measure iff: µ ( ∅ ) = 0 µ ( X ) = 1 For all disjoint A , B ∈ B , µ ( A ∪ B ) = µ ( A ) + µ ( B ) For this talk, assume all measures are finitely additive probability measures. Definition If M is a model, we call a finitely additive probability measure on the boolean algebra of all definable subsets of M n a Keisler measure . Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 15 / 38
Finitely Approximated Measures Let X be a set, B ⊆ P ( X ) be a boolean algebra. Definition For any finite sequence p in X of length n ≥ 1, let Fr p denote the frequency measure determined by p ; i.e., for B ∈ B , we have n Fr p ( B ) = 1 � 1 B ( p i ) . n i =1 Let µ be a measure on B . Definition For F ⊆ B , we say µ is finitely approximated (fap) on F iff: for all ε > 0, there is an ε -approximation p ∈ X such that for all A ∈ F | µ ( A ) − Fr p ( A ) | < ε. Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 16 / 38
E -Definable Sets Let E ⊆ V 1 × · · · × V k and I ⊆ [ k ]. Definition A subset of V I is E-definable over D ⊆ V I c iff: it is a boolean combination of sets of the form E b for b ∈ D . We use B E ( D ) to denote the boolean algebra of all such sets. If D is finite, we use A E ( D ) to denote the atoms in B E ( D ). Roland Walker (UIC) Definable Regularity for NIP Relations April 19, 2017 17 / 38
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