Property testing and hypergraph regularity lemmas Mathias Schacht - PowerPoint PPT Presentation

❋♦r❜ ✐♥❞ ★ Property testing Definition A graph property P is testable with one-sided error if for every η > 0 there exists a constant q = q ( P , η ) and a randomized algorithm A which does the following: For a given graph G the algorithm A can query some oracle q times whether a pair of vertices in V ( G ) spans an edge in G or not and outputs G ∈ P with probability 1 if G ∈ P , G �∈ P with probability at least 2 / 3 if G is η -far from P , and no guarantees otherwise. Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

★ Property testing Definition A graph property P is testable with one-sided error if for every η > 0 there exists a constant q = q ( P , η ) and a randomized algorithm A which does the following: For a given graph G the algorithm A can query some oracle q times whether a pair of vertices in V ( G ) spans an edge in G or not and outputs G ∈ P with probability 1 if G ∈ P , G �∈ P with probability at least 2 / 3 if G is η -far from P , and no guarantees otherwise. Tester G is η -far from P = ❋♦r❜ ✐♥❞ ( F ) Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Property testing Definition A graph property P is testable with one-sided error if for every η > 0 there exists a constant q = q ( P , η ) and a randomized algorithm A which does the following: For a given graph G the algorithm A can query some oracle q times whether a pair of vertices in V ( G ) spans an edge in G or not and outputs G ∈ P with probability 1 if G ∈ P , G �∈ P with probability at least 2 / 3 if G is η -far from P , and no guarantees otherwise. Tester G is η -far from P = ❋♦r❜ ✐♥❞ ( F ) ⇒ ★ { F ≤ G } ≥ cn v F for some F ∈ F with v F ≤ L Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Property testing Definition A graph property P is testable with one-sided error if for every η > 0 there exists a constant q = q ( P , η ) and a randomized algorithm A which does the following: For a given graph G the algorithm A can query some oracle q times whether a pair of vertices in V ( G ) spans an edge in G or not and outputs G ∈ P with probability 1 if G ∈ P , G �∈ P with probability at least 2 / 3 if G is η -far from P , and no guarantees otherwise. Tester G is η -far from P = ❋♦r❜ ✐♥❞ ( F ) ⇒ ★ { F ≤ G } ≥ cn v F for some F ∈ F with v F ≤ L ⇒ “easily” detectable by random sampling; q ∼ f ( L / c ) Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

❋♦r❜ ✐♥❞ ★ ❋♦r❜ ✐♥❞ Proof — Idea Fact If the Theorem fails for F and η > 0, then there exists a sequence of graphs ( G ℓ ) ℓ ∈ N with n ℓ = | V ( G ℓ ) | → ∞ such that for every ℓ ∈ N Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

★ ❋♦r❜ ✐♥❞ Proof — Idea Fact If the Theorem fails for F and η > 0, then there exists a sequence of graphs ( G ℓ ) ℓ ∈ N with n ℓ = | V ( G ℓ ) | → ∞ such that for every ℓ ∈ N (i) G ℓ is η -far from ❋♦r❜ ✐♥❞ ( F ) Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

★ ❋♦r❜ ✐♥❞ Proof — Idea Fact If the Theorem fails for F and η > 0, then there exists a sequence of graphs ( G ℓ ) ℓ ∈ N with n ℓ = | V ( G ℓ ) | → ∞ such that for every ℓ ∈ N (i) G ℓ is η -far from ❋♦r❜ ✐♥❞ ( F ) and (ii) every F ∈ F with v F ≤ ℓ Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

❋♦r❜ ✐♥❞ Proof — Idea Fact If the Theorem fails for F and η > 0, then there exists a sequence of graphs ( G ℓ ) ℓ ∈ N with n ℓ = | V ( G ℓ ) | → ∞ such that for every ℓ ∈ N (i) G ℓ is η -far from ❋♦r❜ ✐♥❞ ( F ) and (ii) every F ∈ F with v F ≤ ℓ ★ { F ≤ G ℓ } ≤ 1 ℓ n v F . Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

❋♦r❜ ✐♥❞ Proof — Idea Fact If the Theorem fails for F and η > 0, then there exists a sequence of graphs ( G ℓ ) ℓ ∈ N with n ℓ = | V ( G ℓ ) | → ∞ such that for every ℓ ∈ N (i) G ℓ is η -far from ❋♦r❜ ✐♥❞ ( F ) and (ii) every F ∈ F with v F ≤ ℓ ★ { F ≤ G ℓ } ≤ 1 ℓ n v F . Idea ( G ℓ ) ℓ ∈ N contains convergent subsequence ( G ′ ℓ ) ℓ ∈ N with limit R Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

❋♦r❜ ✐♥❞ Proof — Idea Fact If the Theorem fails for F and η > 0, then there exists a sequence of graphs ( G ℓ ) ℓ ∈ N with n ℓ = | V ( G ℓ ) | → ∞ such that for every ℓ ∈ N (i) G ℓ is η -far from ❋♦r❜ ✐♥❞ ( F ) and (ii) every F ∈ F with v F ≤ ℓ ★ { F ≤ G ℓ } ≤ 1 ℓ n v F . Idea ( G ℓ ) ℓ ∈ N contains convergent subsequence ( G ′ ℓ ) ℓ ∈ N with limit R R must be ind. F -free, since t ( F , G ℓ ) → 0 for all F ∈ F Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Idea Fact If the Theorem fails for F and η > 0, then there exists a sequence of graphs ( G ℓ ) ℓ ∈ N with n ℓ = | V ( G ℓ ) | → ∞ such that for every ℓ ∈ N (i) G ℓ is η -far from ❋♦r❜ ✐♥❞ ( F ) and (ii) every F ∈ F with v F ≤ ℓ ★ { F ≤ G ℓ } ≤ 1 ℓ n v F . Idea ( G ℓ ) ℓ ∈ N contains convergent subsequence ( G ′ ℓ ) ℓ ∈ N with limit R R must be ind. F -free, since t ( F , G ℓ ) → 0 for all F ∈ F ⇒ “ R has P = ❋♦r❜ ✐♥❞ ( F ) ” Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Idea Fact If the Theorem fails for F and η > 0, then there exists a sequence of graphs ( G ℓ ) ℓ ∈ N with n ℓ = | V ( G ℓ ) | → ∞ such that for every ℓ ∈ N (i) G ℓ is η -far from ❋♦r❜ ✐♥❞ ( F ) and (ii) every F ∈ F with v F ≤ ℓ ★ { F ≤ G ℓ } ≤ 1 ℓ n v F . Idea ( G ℓ ) ℓ ∈ N contains convergent subsequence ( G ′ ℓ ) ℓ ∈ N with limit R R must be ind. F -free, since t ( F , G ℓ ) → 0 for all F ∈ F ⇒ “ R has P = ❋♦r❜ ✐♥❞ ( F ) ” But all G ℓ where η -far from P so R should be η -far from P Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Idea Fact If the Theorem fails for F and η > 0, then there exists a sequence of graphs ( G ℓ ) ℓ ∈ N with n ℓ = | V ( G ℓ ) | → ∞ such that for every ℓ ∈ N (i) G ℓ is η -far from ❋♦r❜ ✐♥❞ ( F ) and (ii) every F ∈ F with v F ≤ ℓ ★ { F ≤ G ℓ } ≤ 1 ℓ n v F . Idea ( G ℓ ) ℓ ∈ N contains convergent subsequence ( G ′ ℓ ) ℓ ∈ N with limit R R must be ind. F -free, since t ( F , G ℓ ) → 0 for all F ∈ F ⇒ “ R has P = ❋♦r❜ ✐♥❞ ( F ) ” But all G ℓ where η -far from P so R should be η -far from P � Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

✭ ✮ ✭ ✮ ❞ ❞ Limits of graph sequences Theorem (Lov´ asz & Szegedy ’06) For every infinite sequence of graphs ( G ℓ ) ℓ ∈ N there exist an infinite subsequence ( G ′ ℓ ) ℓ ∈ N , a sequence of symmetric, measurable functions ( R ℓ ) ℓ ∈ N with R ℓ : [ 0 , 1 ] 2 → [ 0 , 1 ] and a symmetric, measurable function R : [ 0 , 1 ] 2 → [ 0 , 1 ] such that ✭ i ✮ t ✐♥❞ ( F , R ) = lim ℓ →∞ t ✐♥❞ ( F , G ′ ℓ ) for all graphs F Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Limits of graph sequences Theorem (Lov´ asz & Szegedy ’06) For every infinite sequence of graphs ( G ℓ ) ℓ ∈ N there exist an infinite subsequence ( G ′ ℓ ) ℓ ∈ N , a sequence of symmetric, measurable functions ( R ℓ ) ℓ ∈ N with R ℓ : [ 0 , 1 ] 2 → [ 0 , 1 ] and a symmetric, measurable function R : [ 0 , 1 ] 2 → [ 0 , 1 ] such that ✭ i ✮ t ✐♥❞ ( F , R ) = lim ℓ →∞ t ✐♥❞ ( F , G ′ ℓ ) for all graphs F ✭ ii ✮ lim ℓ →∞ d ( G ′ ℓ , R ℓ ) = 0 , and ✭ iii ✮ lim ℓ →∞ d ( R ℓ , R ) = 0 , where for two functions A, B : [ 0 , 1 ] 2 → [ 0 , 1 ] we set � d ( A , B ) sup | A ( x , y ) − B ( x , y ) | ❞ x ❞ y . U ⊆ [ 0 , 1 ] U × U Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Limits of graph sequences Theorem (Lov´ asz & Szegedy ’06) For every infinite sequence of graphs ( G ℓ ) ℓ ∈ N there exist an infinite subsequence ( G ′ ℓ ) ℓ ∈ N , a sequence of symmetric, measurable functions ( R ℓ ) ℓ ∈ N with R ℓ : [ 0 , 1 ] 2 → [ 0 , 1 ] and a symmetric, measurable function R : [ 0 , 1 ] 2 → [ 0 , 1 ] such that ✭ i ✮ t ✐♥❞ ( F , R ) = lim ℓ →∞ t ✐♥❞ ( F , G ′ ℓ ) for all graphs F ✭ ii ✮ lim ℓ →∞ d ( G ′ ℓ , R ℓ ) = 0 , and ✭ iii ✮ lim ℓ →∞ d ( R ℓ , R ) = 0 , where for two functions A, B : [ 0 , 1 ] 2 → [ 0 , 1 ] we set � d ( A , B ) sup | A ( x , y ) − B ( x , y ) | ❞ x ❞ y . U ⊆ [ 0 , 1 ] U × U Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Sketch Proof. let n be sufficiently large and split [0,1] into n intervals Z 1 , . . . , Z n of length 1 / n Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Sketch Proof. let n be sufficiently large and split [0,1] into n intervals Z 1 , . . . , Z n of length 1 / n pick z i ∈ Z i uniformly at random and define R z and H z n  1 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 1 ,   R z / H z n ( x , y ) = 0 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 0 ,   R ( x , y ) / G ′ n ( x , y ) otherwise . Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Sketch Proof. let n be sufficiently large and split [0,1] into n intervals Z 1 , . . . , Z n of length 1 / n pick z i ∈ Z i uniformly at random and define R z and H z n  1 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 1 ,   R z / H z n ( x , y ) = 0 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 0 ,   R ( x , y ) / G ′ n ( x , y ) otherwise . Fact 1 With prob. 1 we have t ( F , H z n ) = 0 for all F ∈ F Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Sketch Proof. let n be sufficiently large and split [0,1] into n intervals Z 1 , . . . , Z n of length 1 / n pick z i ∈ Z i uniformly at random and define R z and H z n  1 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 1 ,   R z / H z n ( x , y ) = 0 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 0 ,   R ( x , y ) / G ′ n ( x , y ) otherwise . Fact 1 With prob. 1 we have t ( F , H z n ) = 0 for all F ∈ F ⇒ H z n ∈ P ⋆ Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Sketch Proof. let n be sufficiently large and split [0,1] into n intervals Z 1 , . . . , Z n of length 1 / n pick z i ∈ Z i uniformly at random and define R z and H z n  1 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 1 ,   R z / H z n ( x , y ) = 0 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 0 ,   R ( x , y ) / G ′ n ( x , y ) otherwise . Fact 1 With prob. 1 we have t ( F , H z n ) = 0 for all F ∈ F ⇒ H z n ∈ P ⋆ Fact 2 W.h.p. d ( R z , R ) = o ( 1 ) . Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Sketch Proof. let n be sufficiently large and split [0,1] into n intervals Z 1 , . . . , Z n of length 1 / n pick z i ∈ Z i uniformly at random and define R z and H z n  1 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 1 ,   R z / H z n ( x , y ) = 0 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 0 ,   R ( x , y ) / G ′ n ( x , y ) otherwise . Fact 1 With prob. 1 we have t ( F , H z n ) = 0 for all F ∈ F ⇒ H z n ∈ P ⋆ Fact 2 W.h.p. d ( R z , R ) = o ( 1 ) . ⇒ d ( H z n , R z ) Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Sketch Proof. let n be sufficiently large and split [0,1] into n intervals Z 1 , . . . , Z n of length 1 / n pick z i ∈ Z i uniformly at random and define R z and H z n  1 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 1 ,   R z / H z n ( x , y ) = 0 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 0 ,   R ( x , y ) / G ′ n ( x , y ) otherwise . Fact 1 With prob. 1 we have t ( F , H z n ) = 0 for all F ∈ F ⇒ H z n ∈ P ⋆ Fact 2 W.h.p. d ( R z , R ) = o ( 1 ) . ⇒ d ( H z n , R z ) ≤ d ( G ′ n , R ) Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Sketch Proof. let n be sufficiently large and split [0,1] into n intervals Z 1 , . . . , Z n of length 1 / n pick z i ∈ Z i uniformly at random and define R z and H z n  1 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 1 ,   R z / H z n ( x , y ) = 0 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 0 ,   R ( x , y ) / G ′ n ( x , y ) otherwise . Fact 1 With prob. 1 we have t ( F , H z n ) = 0 for all F ∈ F ⇒ H z n ∈ P ⋆ Fact 2 W.h.p. d ( R z , R ) = o ( 1 ) . ⇒ d ( H z n , R z ) ≤ d ( G ′ n , R ) ≤ d ( G ′ n , R n ) + d ( R n , R ) Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Sketch Proof. let n be sufficiently large and split [0,1] into n intervals Z 1 , . . . , Z n of length 1 / n pick z i ∈ Z i uniformly at random and define R z and H z n  1 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 1 ,   R z / H z n ( x , y ) = 0 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 0 ,   R ( x , y ) / G ′ n ( x , y ) otherwise . Fact 1 With prob. 1 we have t ( F , H z n ) = 0 for all F ∈ F ⇒ H z n ∈ P ⋆ Fact 2 W.h.p. d ( R z , R ) = o ( 1 ) . ⇒ d ( H z n , R z ) ≤ d ( G ′ n , R ) ≤ d ( G ′ n , R n ) + d ( R n , R ) = o ( 1 ) Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Sketch Proof. let n be sufficiently large and split [0,1] into n intervals Z 1 , . . . , Z n of length 1 / n pick z i ∈ Z i uniformly at random and define R z and H z n  1 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 1 ,   R z / H z n ( x , y ) = 0 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 0 ,   R ( x , y ) / G ′ n ( x , y ) otherwise . Fact 1 With prob. 1 we have t ( F , H z n ) = 0 for all F ∈ F ⇒ H z n ∈ P ⋆ Fact 2 W.h.p. d ( R z , R ) = o ( 1 ) . ⇒ d ( H z n , R z ) ≤ d ( G ′ n , R ) ≤ d ( G ′ n , R n ) + d ( R n , R ) = o ( 1 ) ⇒ d ( H z n , G ′ n ) Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Sketch Proof. let n be sufficiently large and split [0,1] into n intervals Z 1 , . . . , Z n of length 1 / n pick z i ∈ Z i uniformly at random and define R z and H z n  1 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 1 ,   R z / H z n ( x , y ) = 0 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 0 ,   R ( x , y ) / G ′ n ( x , y ) otherwise . Fact 1 With prob. 1 we have t ( F , H z n ) = 0 for all F ∈ F ⇒ H z n ∈ P ⋆ Fact 2 W.h.p. d ( R z , R ) = o ( 1 ) . ⇒ d ( H z n , R z ) ≤ d ( G ′ n , R ) ≤ d ( G ′ n , R n ) + d ( R n , R ) = o ( 1 ) ⇒ d ( H z n ) ≤ d ( H z , R z ) + d ( R z , R ) + d ( R , G ′ n , G ′ n ) Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Sketch Proof. let n be sufficiently large and split [0,1] into n intervals Z 1 , . . . , Z n of length 1 / n pick z i ∈ Z i uniformly at random and define R z and H z n  1 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 1 ,   R z / H z n ( x , y ) = 0 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 0 ,   R ( x , y ) / G ′ n ( x , y ) otherwise . Fact 1 With prob. 1 we have t ( F , H z n ) = 0 for all F ∈ F ⇒ H z n ∈ P ⋆ Fact 2 W.h.p. d ( R z , R ) = o ( 1 ) . ⇒ d ( H z n , R z ) ≤ d ( G ′ n , R ) ≤ d ( G ′ n , R n ) + d ( R n , R ) = o ( 1 ) ⇒ d ( H z n ) ≤ d ( H z , R z ) + d ( R z , R ) + d ( R , G ′ n , G ′ n ) = o ( 1 ) Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Sketch Proof. let n be sufficiently large and split [0,1] into n intervals Z 1 , . . . , Z n of length 1 / n pick z i ∈ Z i uniformly at random and define R z and H z n  1 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 1 ,   R z / H z n ( x , y ) = 0 if ( x , y ) ∈ Z i × Z j and R ( z i , z j ) = 0 ,   R ( x , y ) / G ′ n ( x , y ) otherwise . Fact 1 With prob. 1 we have t ( F , H z n ) = 0 for all F ∈ F ⇒ H z n ∈ P ⋆ Fact 2 W.h.p. d ( R z , R ) = o ( 1 ) . ⇒ d ( H z n , R z ) ≤ d ( G ′ n , R ) ≤ d ( G ′ n , R n ) + d ( R n , R ) = o ( 1 ) ⇒ d ( H z n ) ≤ d ( H z , R z ) + d ( R z , R ) + d ( R , G ′ n , G ′ n ) = o ( 1 ) ⇒ H z is η -close to G ′ n and � due to ⋆ . Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Outline 1 Property testing for graphs 2 Quasi-random hypergraphs Quasi-random graphs Three possible extensions The “right” extension Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

★ Quasi-random properties of graphs Definition ( disc ) A graph G = ( V , E ) of density d satisfies disc if � | U | � | = o ( n 2 ) for all U ⊆ V . | e ( U ) − d 2 Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

★ Quasi-random properties of graphs Definition ( disc ) A graph G = ( V , E ) of density d satisfies disc if � | U | � | = o ( n 2 ) for all U ⊆ V . | e ( U ) − d 2 Definition ( dev ) A graph G = ( V , E ) of density d satisfies dev if � � � � g ( u i , v j ) = o ( n 4 ) , u 0 , u 1 ∈ V n v 0 , v 1 ∈ V n i ∈ { 0 , 1 } j ∈ { 0 , 1 } where g ( u , v ) = 1 − d if { u , v } ∈ E and g ( u , v ) = − d if { u , v } �∈ E . Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random properties of graphs Definition ( disc ) A graph G = ( V , E ) of density d satisfies disc if � | U | � | = o ( n 2 ) for all U ⊆ V . | e ( U ) − d 2 Definition ( dev ) A graph G = ( V , E ) of density d satisfies dev if � � � � g ( u i , v j ) = o ( n 4 ) , u 0 , u 1 ∈ V n v 0 , v 1 ∈ V n i ∈ { 0 , 1 } j ∈ { 0 , 1 } where g ( u , v ) = 1 − d if { u , v } ∈ E and g ( u , v ) = − d if { u , v } �∈ E . Definition ( cycle ) A graph G = ( V , E ) of density d satisfies cycle if ★ { C 4 ⊆ G } ≤ d 4 n 4 + o ( n 4 ) . Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random graphs Theorem (Chung, Graham & Wilson ’89 and others) The properties disc , dev , and cycle are equivalent for every d ∈ ( 0 , 1 ] . Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random graphs Theorem (Chung, Graham & Wilson ’89 and others) The properties disc , dev , and cycle are equivalent for every d ∈ ( 0 , 1 ] . Remarks several other equivalent properties are known Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random graphs Theorem (Chung, Graham & Wilson ’89 and others) The properties disc , dev , and cycle are equivalent for every d ∈ ( 0 , 1 ] . Remarks several other equivalent properties are known equivalence also holds for bipartite versions of disc , dev , and cycle Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random graphs Theorem (Chung, Graham & Wilson ’89 and others) The properties disc , dev , and cycle are equivalent for every d ∈ ( 0 , 1 ] . Remarks several other equivalent properties are known equivalence also holds for bipartite versions of disc , dev , and cycle bipartite version of disc corresponds to the concept of ε -regular pairs of Szemer´ edi’s regularity lemma Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random graphs Theorem (Chung, Graham & Wilson ’89 and others) The properties disc , dev , and cycle are equivalent for every d ∈ ( 0 , 1 ] . Remarks several other equivalent properties are known equivalence also holds for bipartite versions of disc , dev , and cycle bipartite version of disc corresponds to the concept of ε -regular pairs of Szemer´ edi’s regularity lemma Question How do we generalize those concepts for hypergraphs? Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

★ ★ Quasi-random hypergraphs: First attempt Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

★ ★ Quasi-random hypergraphs: First attempt Definition ( weak-disc ) A 3-uniform hypergraph H = ( V , E ) of density d satisfies weak-disc if � | U | � | = o ( n 3 ) for all U ⊆ V . | e ( U ) − d 3 Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

★ Quasi-random hypergraphs: First attempt Definition ( weak-disc ) A 3-uniform hypergraph H = ( V , E ) of density d satisfies weak-disc if � | U | � | = o ( n 3 ) for all U ⊆ V . | e ( U ) − d 3 Definition ( oct ) A 3-uniform hypergraph H = ( V , E ) of density d satisfies oct if 2 , 2 , 2 ⊆ H } ≤ d 8 n 6 + o ( n 6 ) . ★ { K ( 3 ) Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

★ Quasi-random hypergraphs: First attempt Definition ( weak-disc ) A 3-uniform hypergraph H = ( V , E ) of density d satisfies weak-disc if � | U | � | = o ( n 3 ) for all U ⊆ V . | e ( U ) − d 3 Definition ( oct ) A 3-uniform hypergraph H = ( V , E ) of density d satisfies oct if 2 , 2 , 2 ⊆ H } ≤ d 8 n 6 + o ( n 6 ) . ★ { K ( 3 ) Fact There exists a 3-uniform hypergraph H = ( V , E ) of density 1 / 8 ± o ( 1 ) such that Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

★ Quasi-random hypergraphs: First attempt Definition ( weak-disc ) A 3-uniform hypergraph H = ( V , E ) of density d satisfies weak-disc if � | U | � | = o ( n 3 ) for all U ⊆ V . | e ( U ) − d 3 Definition ( oct ) A 3-uniform hypergraph H = ( V , E ) of density d satisfies oct if 2 , 2 , 2 ⊆ H } ≤ d 8 n 6 + o ( n 6 ) . ★ { K ( 3 ) Fact There exists a 3-uniform hypergraph H = ( V , E ) of density 1 / 8 ± o ( 1 ) such that H satisfies weak-disc Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: First attempt Definition ( weak-disc ) A 3-uniform hypergraph H = ( V , E ) of density d satisfies weak-disc if � | U | � | = o ( n 3 ) for all U ⊆ V . | e ( U ) − d 3 Definition ( oct ) A 3-uniform hypergraph H = ( V , E ) of density d satisfies oct if 2 , 2 , 2 ⊆ H } ≤ d 8 n 6 + o ( n 6 ) . ★ { K ( 3 ) Fact There exists a 3-uniform hypergraph H = ( V , E ) of density 1 / 8 ± o ( 1 ) such that H satisfies weak-disc , but 2 ) 12 n 6 = ( 1 − o ( 1 ))( 1 8 ) 4 n 6 > ( 1 − o ( 1 ))( 1 ★ { K ( 3 ) 2 , 2 , 2 ⊆ H } ≥ ( 1 − o ( 1 ))( 1 8 ) 8 n 6 . Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Second attempt Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Second attempt Definition ( disc ) A 3-uniform hypergraph H = ( V , E ) of density d satisfies disc if � = o ( n 3 ) for all graphs G with vertex set V . � � � | E ∩ K 3 ( G ) | − d | K 3 ( G ) | Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Second attempt Definition ( disc ) A 3-uniform hypergraph H = ( V , E ) of density d satisfies disc if � = o ( n 3 ) for all graphs G with vertex set V . � � � | E ∩ K 3 ( G ) | − d | K 3 ( G ) | Definition ( dev ) A 3-uniform hypergraph H = ( V , E ) of density d satisfies dev if � � � � h ( u i , v j , w k ) = o ( n 6 ) , u 0 , u 1 v 0 , v 1 w 0 , w 1 i , j , k ∈ { 0 , 1 } where h ( u , v , w ) = 1 − d if { u , v , w } ∈ E and − d otherwise. Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Second attempt Definition ( disc ) A 3-uniform hypergraph H = ( V , E ) of density d satisfies disc if � = o ( n 3 ) for all graphs G with vertex set V . � � � | E ∩ K 3 ( G ) | − d | K 3 ( G ) | Definition ( dev ) A 3-uniform hypergraph H = ( V , E ) of density d satisfies dev if � � � � h ( u i , v j , w k ) = o ( n 6 ) , u 0 , u 1 v 0 , v 1 w 0 , w 1 i , j , k ∈ { 0 , 1 } where h ( u , v , w ) = 1 − d if { u , v , w } ∈ E and − d otherwise. Remarks disc , dev , and oct are equivalent (Chung & Graham 1990) Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Second attempt Definition ( disc ) A 3-uniform hypergraph H = ( V , E ) of density d satisfies disc if � = o ( n 3 ) for all graphs G with vertex set V . � � � | E ∩ K 3 ( G ) | − d | K 3 ( G ) | Definition ( dev ) A 3-uniform hypergraph H = ( V , E ) of density d satisfies dev if � � � � h ( u i , v j , w k ) = o ( n 6 ) , u 0 , u 1 v 0 , v 1 w 0 , w 1 i , j , k ∈ { 0 , 1 } where h ( u , v , w ) = 1 − d if { u , v , w } ∈ E and − d otherwise. Remarks disc , dev , and oct are equivalent (Chung & Graham 1990) but a regularity lemma for this concept of disc requires partition of the edge set of the complete graph on the same vertex set Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

★ Quasi-random hypergraphs: Third attempt Set-Up a graph G = ( V , E G ) of density d 2 satisfying disc / dev / oct Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

★ Quasi-random hypergraphs: Third attempt Set-Up a graph G = ( V , E G ) of density d 2 satisfying disc / dev / oct a 3-uniform hypergraph H = ( V , E H ) ⊆K 3 ( G ) with relative density d 3 Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

★ Quasi-random hypergraphs: Third attempt Set-Up a graph G = ( V , E G ) of density d 2 satisfying disc / dev / oct a 3-uniform hypergraph H = ( V , E H ) ⊆K 3 ( G ) with relative density d 3 Definition ( disc ) ( G , H ) satisfies disc if � = o ( n 3 ) for all subgraphs G ′ ⊆ G . � � | E ∩ K 3 ( G ′ ) | − d 3 | K 3 ( G ′ ) | � Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Third attempt Set-Up a graph G = ( V , E G ) of density d 2 satisfying disc / dev / oct a 3-uniform hypergraph H = ( V , E H ) ⊆K 3 ( G ) with relative density d 3 Definition ( disc ) ( G , H ) satisfies disc if � = o ( n 3 ) for all subgraphs G ′ ⊆ G . � | E ∩ K 3 ( G ′ ) | − d 3 | K 3 ( G ′ ) | � � Definition ( oct ) ( G , H ) satisfies oct if ★ { K ( 3 ) 2 n 6 + o ( n 6 ) . 2 , 2 , 2 ⊆ H } ≤ d 8 3 d 12 Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Third attempt (cont’d) Set-Up a graph G = ( V , E G ) of density d 2 satisfying disc / dev / oct a 3-uniform hypergraph H = ( V , E H ) ⊆K 3 ( G ) with relative density d 3 Definition ( dev ) ( G , H ) satisfies dev if Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Third attempt (cont’d) Set-Up a graph G = ( V , E G ) of density d 2 satisfying disc / dev / oct a 3-uniform hypergraph H = ( V , E H ) ⊆K 3 ( G ) with relative density d 3 Definition ( dev ) ( G , H ) satisfies dev if � � � � h ( u i , v j , w k ) = o ( n 6 ) , u 0 , u 1 v 0 , v 1 w 0 , w 1 i , j , k ∈ { 0 , 1 }   1 − d if { u , v , w } ∈ E H ,   where h ( u , v , w ) = − d if { u , v , w } ∈ K 3 ( G ) \ E H ,    0 otherwise . Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Third attempt (cont’d) Set-Up a graph G = ( V , E G ) of density d 2 satisfying disc / dev / oct a 3-uniform hypergraph H = ( V , E H ) ⊆K 3 ( G ) with relative density d 3 Remarks disc , dev , and oct are equivalent (similar to Chung & Graham 1990) Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Third attempt (cont’d) Set-Up a graph G = ( V , E G ) of density d 2 satisfying disc / dev / oct a 3-uniform hypergraph H = ( V , E H ) ⊆K 3 ( G ) with relative density d 3 Remarks disc , dev , and oct are equivalent (similar to Chung & Graham 1990) but the error-terms are only “useful” if ε ≪ d 2 , i.e., Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Third attempt (cont’d) Set-Up a graph G = ( V , E G ) of density d 2 satisfying disc / dev / oct a 3-uniform hypergraph H = ( V , E H ) ⊆K 3 ( G ) with relative density d 3 Remarks disc , dev , and oct are equivalent (similar to Chung & Graham 1990) but the error-terms are only “useful” if ε ≪ d 2 , i.e., disc : o ( n 3 ) ≪ d 3 d 3 2 n 3 Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Third attempt (cont’d) Set-Up a graph G = ( V , E G ) of density d 2 satisfying disc / dev / oct a 3-uniform hypergraph H = ( V , E H ) ⊆K 3 ( G ) with relative density d 3 Remarks disc , dev , and oct are equivalent (similar to Chung & Graham 1990) but the error-terms are only “useful” if ε ≪ d 2 , i.e., disc : o ( n 3 ) ≪ d 3 d 3 2 n 3 oct : o ( n 6 ) ≪ d 8 3 d 12 2 n 6 dev : o ( n 6 ) ≪ d 8 3 d 12 2 n 6 Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Third attempt (cont’d) Set-Up a graph G = ( V , E G ) of density d 2 satisfying disc / dev / oct a 3-uniform hypergraph H = ( V , E H ) ⊆K 3 ( G ) with relative density d 3 Remarks disc , dev , and oct are equivalent (similar to Chung & Graham 1990) but the error-terms are only “useful” if ε ≪ d 2 , i.e., disc : o ( n 3 ) ≪ d 3 d 3 2 n 3 oct : o ( n 6 ) ≪ d 8 3 d 12 2 n 6 dev : o ( n 6 ) ≪ d 8 3 d 12 2 n 6 Bad Fact There is noregularity lemma for hypergraphs possible such that the “quasi-randomness” of the hypergraph (i.e., ε ) “beats” the density of the underlying graphs (i.e., d 2 ) provided by such a regularity lemma. Similar as ε ≫ 1 / t in Szemer´ edi’s regularity lemma. Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

★ Quasi-random hypergraphs: Final attempt Set-Up a graph G = ( V , E G ) of density d 2 satisfying disc ( ε 2 ) / dev ( ε 2 ) / oct ( ε 2 ) a 3-uniform hypergraph H = ( V , E H ) ⊆K 3 ( G ) with relative density d 3 Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

★ Quasi-random hypergraphs: Final attempt Set-Up a graph G = ( V , E G ) of density d 2 satisfying disc ( ε 2 ) / dev ( ε 2 ) / oct ( ε 2 ) a 3-uniform hypergraph H = ( V , E H ) ⊆K 3 ( G ) with relative density d 3 d 3 ≫ ε 3 ≫ d 2 ≫ ε 2 Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

★ Quasi-random hypergraphs: Final attempt Set-Up a graph G = ( V , E G ) of density d 2 satisfying disc ( ε 2 ) / dev ( ε 2 ) / oct ( ε 2 ) a 3-uniform hypergraph H = ( V , E H ) ⊆K 3 ( G ) with relative density d 3 d 3 ≫ ε 3 ≫ d 2 ≫ ε 2 Definition ( disc ) ( G , H ) satisfies disc ( ε 3 ) if 2 n 3 for all subgraphs G ′ ⊆ G . � ≤ ε 3 d 3 � � � | E ∩ K 3 ( G ′ ) | − d 3 | K 3 ( G ′ ) | Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Final attempt Set-Up a graph G = ( V , E G ) of density d 2 satisfying disc ( ε 2 ) / dev ( ε 2 ) / oct ( ε 2 ) a 3-uniform hypergraph H = ( V , E H ) ⊆K 3 ( G ) with relative density d 3 d 3 ≫ ε 3 ≫ d 2 ≫ ε 2 Definition ( disc ) ( G , H ) satisfies disc ( ε 3 ) if 2 n 3 for all subgraphs G ′ ⊆ G . � ≤ ε 3 d 3 � � � | E ∩ K 3 ( G ′ ) | − d 3 | K 3 ( G ′ ) | Definition ( oct ) ( G , H ) satisfies oct ( ε 3 ) if 2 n 6 + ε 3 d 12 2 n 6 . ★ { K ( 3 ) 2 , 2 , 2 ⊆ H } ≤ d 8 3 d 12 Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Final attempt (cont’d) Set-Up a graph G = ( V , E G ) of density d 2 satisfying disc ( ε 2 ) / dev ( ε 2 ) / oct ( ε 2 ) a 3-uniform hypergraph H = ( V , E H ) ⊆K 3 ( G ) with relative density d 3 d 3 ≫ ε 3 ≫ d 2 ≫ ε 2 Definition ( dev ) ( G , H ) satisfies dev ( ε 3 ) if Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Final attempt (cont’d) Set-Up a graph G = ( V , E G ) of density d 2 satisfying disc ( ε 2 ) / dev ( ε 2 ) / oct ( ε 2 ) a 3-uniform hypergraph H = ( V , E H ) ⊆K 3 ( G ) with relative density d 3 d 3 ≫ ε 3 ≫ d 2 ≫ ε 2 Definition ( dev ) ( G , H ) satisfies dev ( ε 3 ) if � � � � 2 n 6 , h ( u i , v j , w k ) ≤ ε 3 d 12 u 0 , u 1 v 0 , v 1 w 0 , w 1 i , j , k ∈ { 0 , 1 }   1 − d if { u , v , w } ∈ E H ,   where h ( u , v , w ) = − d if { u , v , w } ∈ K 3 ( G ) \ E H ,    0 otherwise . Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Remarks Earlier Results For those concepts of quasi-random hypergraphs there exist regularity lemmas which: partition the vertex set and the set of pairs such that “most” block satisfy disc / dev / oct Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Remarks Earlier Results For those concepts of quasi-random hypergraphs there exist regularity lemmas which: partition the vertex set and the set of pairs such that “most” block satisfy disc / dev / oct Regularity Lemmas due to Frankl–R¨ odl ( disc ), Gowers ( dev ), and Haxel–Nagle–R¨ odl ( oct ) Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Remarks Earlier Results For those concepts of quasi-random hypergraphs there exist regularity lemmas which: partition the vertex set and the set of pairs such that “most” block satisfy disc / dev / oct Regularity Lemmas due to Frankl–R¨ odl ( disc ), Gowers ( dev ), and Haxel–Nagle–R¨ odl ( oct ) corresponding Counting Lemmas are known for dev (general k ), oct ( k = 3), stronger versions of disc (general k ) Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Remarks Earlier Results For those concepts of quasi-random hypergraphs there exist regularity lemmas which: partition the vertex set and the set of pairs such that “most” block satisfy disc / dev / oct Regularity Lemmas due to Frankl–R¨ odl ( disc ), Gowers ( dev ), and Haxel–Nagle–R¨ odl ( oct ) corresponding Counting Lemmas are known for dev (general k ), oct ( k = 3), stronger versions of disc (general k ) algorithmic Regularity Lemma only known for oct ( k = 3) Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

New Result Theorem (Nagle, Poerschke, R¨ odl, S.) disc , dev , and oct are equivalent for 3 -uniform hypergraphs. Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

New Result Theorem (Nagle, Poerschke, R¨ odl, S.) disc , dev , and oct are equivalent for 3 -uniform hypergraphs. I.e., For all d 3 , ε 3 there exists δ 3 such that for all d 2 , ε 2 there exists δ 2 and n 0 such that Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

New Result Theorem (Nagle, Poerschke, R¨ odl, S.) disc , dev , and oct are equivalent for 3 -uniform hypergraphs. I.e., For all d 3 , ε 3 there exists δ 3 such that for all d 2 , ε 2 there exists δ 2 and n 0 such that if ( H , G ) with densities d 3 and d 2 satisfies disc ( δ 2 , δ 3 ) , Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

New Result Theorem (Nagle, Poerschke, R¨ odl, S.) disc , dev , and oct are equivalent for 3 -uniform hypergraphs. I.e., For all d 3 , ε 3 there exists δ 3 such that for all d 2 , ε 2 there exists δ 2 and n 0 such that if ( H , G ) with densities d 3 and d 2 satisfies disc ( δ 2 , δ 3 ) , then ( H , G ) must satisfy dev ( ε 2 , ε 3 ) ; . . . Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

New Result Theorem (Nagle, Poerschke, R¨ odl, S.) disc , dev , and oct are equivalent for 3 -uniform hypergraphs. I.e., For all d 3 , ε 3 there exists δ 3 such that for all d 2 , ε 2 there exists δ 2 and n 0 such that if ( H , G ) with densities d 3 and d 2 satisfies disc ( δ 2 , δ 3 ) , then ( H , G ) must satisfy dev ( ε 2 , ε 3 ) ; . . . Remarks main part of the proof: disc ⇒ dev / oct Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

New Result Theorem (Nagle, Poerschke, R¨ odl, S.) disc , dev , and oct are equivalent for 3 -uniform hypergraphs. I.e., For all d 3 , ε 3 there exists δ 3 such that for all d 2 , ε 2 there exists δ 2 and n 0 such that if ( H , G ) with densities d 3 and d 2 satisfies disc ( δ 2 , δ 3 ) , then ( H , G ) must satisfy dev ( ε 2 , ε 3 ) ; . . . Remarks main part of the proof: disc ⇒ dev / oct proof is based on two applications of the hypergraph regularity lemma Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

New Result Theorem (Nagle, Poerschke, R¨ odl, S.) disc , dev , and oct are equivalent for 3 -uniform hypergraphs. I.e., For all d 3 , ε 3 there exists δ 3 such that for all d 2 , ε 2 there exists δ 2 and n 0 such that if ( H , G ) with densities d 3 and d 2 satisfies disc ( δ 2 , δ 3 ) , then ( H , G ) must satisfy dev ( ε 2 , ε 3 ) ; . . . Remarks main part of the proof: disc ⇒ dev / oct proof is based on two applications of the hypergraph regularity lemma other implication follow from the Counting Lemmas known for dev and oct Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Consequences and concluding remarks Consequences for 3-uniform hypergraphs Counting Lemma for this version of disc Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Consequences and concluding remarks Consequences for 3-uniform hypergraphs Counting Lemma for this version of disc stronger version of disc due to Frankl–R¨ odl is not equivalent to disc / dev / oct Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008