Graphs and limits Mathias Schacht Institut f ur Informatik - PowerPoint PPT Presentation

Graphs and limits Mathias Schacht Institut f¨ ur Informatik Humboldt-Universit¨ at zu Berlin November 2008 Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Outline 1 Regularity lemmas for graphs Frieze-Kannan Lemma Tao’s Lemma Szemer´ edi’s Lemma AFKS-Lemma Counting Lemma and subgraph frequencies Related Lemmas Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Outline 1 Regularity lemmas for graphs Frieze-Kannan Lemma Tao’s Lemma Szemer´ edi’s Lemma AFKS-Lemma Counting Lemma and subgraph frequencies Related Lemmas 2 Limits of graph sequences The limit object Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Introduction History mid-seventies Szemer´ edi discovered the regularity lemma Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Introduction History mid-seventies Szemer´ edi discovered the regularity lemma until late nineties it was further developed and applied by Szemer´ edi et al., R¨ odl et al., Hungarians, and Alon et al. Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Introduction History mid-seventies Szemer´ edi discovered the regularity lemma until late nineties it was further developed and applied by Szemer´ edi et al., R¨ odl et al., Hungarians, and Alon et al. few years ago “different” hypergraph extensions by R¨ odl et al., Gowers, Tao appeared Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Introduction History mid-seventies Szemer´ edi discovered the regularity lemma until late nineties it was further developed and applied by Szemer´ edi et al., R¨ odl et al., Hungarians, and Alon et al. few years ago “different” hypergraph extensions by R¨ odl et al., Gowers, Tao appeared several variants of Szemer´ edi’s lemma and the Lov´ asz-Szegedy limit approach appeared Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Introduction History mid-seventies Szemer´ edi discovered the regularity lemma until late nineties it was further developed and applied by Szemer´ edi et al., R¨ odl et al., Hungarians, and Alon et al. few years ago “different” hypergraph extensions by R¨ odl et al., Gowers, Tao appeared several variants of Szemer´ edi’s lemma and the Lov´ asz-Szegedy limit approach appeared Questions Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Introduction History mid-seventies Szemer´ edi discovered the regularity lemma until late nineties it was further developed and applied by Szemer´ edi et al., R¨ odl et al., Hungarians, and Alon et al. few years ago “different” hypergraph extensions by R¨ odl et al., Gowers, Tao appeared several variants of Szemer´ edi’s lemma and the Lov´ asz-Szegedy limit approach appeared Questions How do these regularity lemmas relate to each other? Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Introduction History mid-seventies Szemer´ edi discovered the regularity lemma until late nineties it was further developed and applied by Szemer´ edi et al., R¨ odl et al., Hungarians, and Alon et al. few years ago “different” hypergraph extensions by R¨ odl et al., Gowers, Tao appeared several variants of Szemer´ edi’s lemma and the Lov´ asz-Szegedy limit approach appeared Questions How do these regularity lemmas relate to each other? Are they different or is it all the same? Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Introduction History mid-seventies Szemer´ edi discovered the regularity lemma until late nineties it was further developed and applied by Szemer´ edi et al., R¨ odl et al., Hungarians, and Alon et al. few years ago “different” hypergraph extensions by R¨ odl et al., Gowers, Tao appeared several variants of Szemer´ edi’s lemma and the Lov´ asz-Szegedy limit approach appeared Questions How do these regularity lemmas relate to each other? Are they different or is it all the same? Which lemma is good for what? Mathias Schacht (HU-Berlin) Graphs and limits November 2008

✭ ✮ ✭ ✮ A simple regularity lemma Theorem (Frieze-Kannan ’99) For every ε > 0 exists T 0 such that for every n-vertex graph G = ( V , E ) there exists a partition V 1 ❴ ∪ . . . ❴ ∪ V t = V satisfying ✭ i ✮ t ≤ T 0 , Mathias Schacht (HU-Berlin) Graphs and limits November 2008

✭ ✮ A simple regularity lemma Theorem (Frieze-Kannan ’99) For every ε > 0 exists T 0 such that for every n-vertex graph G = ( V , E ) there exists a partition V 1 ❴ ∪ . . . ❴ ∪ V t = V satisfying ✭ i ✮ t ≤ T 0 , ✭ ii ✮ | V 1 | ≤ · · · ≤ | V t | ≤ | V 1 | + 1 , Mathias Schacht (HU-Berlin) Graphs and limits November 2008

A simple regularity lemma Theorem (Frieze-Kannan ’99) For every ε > 0 exists T 0 such that for every n-vertex graph G = ( V , E ) there exists a partition V 1 ❴ ∪ . . . ❴ ∪ V t = V satisfying ✭ i ✮ t ≤ T 0 , ✭ ii ✮ | V 1 | ≤ · · · ≤ | V t | ≤ | V 1 | + 1 , and ✭ iii ✮ for every U ⊆ V we have � � � � � ≤ ε n 2 . � � e ( U ) − d ( V i , V j ) | U ∩ V i || U ∩ V j | � � � � 1 ≤ i < j ≤ t � � Mathias Schacht (HU-Berlin) Graphs and limits November 2008

✭ ✮ ✭ ✮ ✭ ✮ ✐♥❞ ✐♥❞ ❴ ❴ ❴ ❴ ✭ ✮ ✭ ✮ Proof of FK-Lemma Definition (Index) Let G = ( V , E ) be a graph and let V = V 1 ❴ ∪ . . . ❴ ∪ V t be a partition of V . We define the index of V by � ✐♥❞ ( V ) = 1 d 2 ( V i , V j ) | V i || V j | . n 2 i < j Mathias Schacht (HU-Berlin) Graphs and limits November 2008

✭ ✮ ✐♥❞ ✐♥❞ ❴ ❴ ❴ ❴ ✭ ✮ ✭ ✮ Proof of FK-Lemma Definition (Index) Let G = ( V , E ) be a graph and let V = V 1 ❴ ∪ . . . ❴ ∪ V t be a partition of V . We define the index of V by � ✐♥❞ ( V ) = 1 d 2 ( V i , V j ) | V i || V j | . n 2 i < j Proof. Iteratively define partitions V 1 , . . . such that ✭ i ✮ and ✭ ii ✮ hold. Mathias Schacht (HU-Berlin) Graphs and limits November 2008

✭ ✮ ✭ ✮ Proof of FK-Lemma Definition (Index) Let G = ( V , E ) be a graph and let V = V 1 ❴ ∪ . . . ❴ ∪ V t be a partition of V . We define the index of V by � ✐♥❞ ( V ) = 1 d 2 ( V i , V j ) | V i || V j | . n 2 i < j Proof. Iteratively define partitions V 1 , . . . such that ✭ i ✮ and ✭ ii ✮ hold. If U does not satisfy ✭ iii ✮ for V i , then ✐♥❞ ( W ) ≥ ✐♥❞ ( V i ) + ε 2 for Q = ( V 1 ∩ U ) ❴ ∪ ( V 1 \ U ) ❴ ∪ . . . ❴ ∪ ( V t i ∩ U ) ❴ ∪ ( V t i \ U ) . Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Proof of FK-Lemma Definition (Index) Let G = ( V , E ) be a graph and let V = V 1 ❴ ∪ . . . ❴ ∪ V t be a partition of V . We define the index of V by � ✐♥❞ ( V ) = 1 d 2 ( V i , V j ) | V i || V j | . n 2 i < j Proof. Iteratively define partitions V 1 , . . . such that ✭ i ✮ and ✭ ii ✮ hold. If U does not satisfy ✭ iii ✮ for V i , then ✐♥❞ ( W ) ≥ ✐♥❞ ( V i ) + ε 2 for Q = ( V 1 ∩ U ) ❴ ∪ ( V 1 \ U ) ❴ ∪ . . . ❴ ∪ ( V t i ∩ U ) ❴ ∪ ( V t i \ U ) . “Massage” W and obtain V i + 1 , which satisfies ✭ i ✮ , ✭ ii ✮ and which “keeps” the index-increment. Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Proof of FK-Lemma Definition (Index) Let G = ( V , E ) be a graph and let V = V 1 ❴ ∪ . . . ❴ ∪ V t be a partition of V . We define the index of V by � ✐♥❞ ( V ) = 1 d 2 ( V i , V j ) | V i || V j | . n 2 i < j Proof. Iteratively define partitions V 1 , . . . such that ✭ i ✮ and ✭ ii ✮ hold. If U does not satisfy ✭ iii ✮ for V i , then ✐♥❞ ( W ) ≥ ✐♥❞ ( V i ) + ε 2 for Q = ( V 1 ∩ U ) ❴ ∪ ( V 1 \ U ) ❴ ∪ . . . ❴ ∪ ( V t i ∩ U ) ❴ ∪ ( V t i \ U ) . “Massage” W and obtain V i + 1 , which satisfies ✭ i ✮ , ✭ ii ✮ and which “keeps” the index-increment. Procedure must end after at most O ( 1 /ε 2 ) iterations. Mathias Schacht (HU-Berlin) Graphs and limits November 2008

❴ ❴ ❴ ❴ The FK-Lemma, revisited Theorem For every ε > 0 , t 0 ∈ N exists T 0 and n 0 such that for every n-vertex graph G = ( V , E ) there exists a partition V 1 ❴ ∪ . . . ❴ ∪ V t = V satisfying ✭ i ✮ t 0 ≤ t ≤ T 0 , ✭ ii ✮ | V i | = n / t, and ✭ iii ✮ for every U ⊆ V we have � � � � � ≤ ε n 2 . � � e ( U ) − d ( V i , V j ) | U ∩ V i || U ∩ V j | � � � � 1 ≤ i < j ≤ t � � Mathias Schacht (HU-Berlin) Graphs and limits November 2008

❴ ❴ ❴ ❴ The FK-Lemma, revisited Theorem For every ε > 0 , t 0 ∈ N exists T 0 and n 0 such that for every n-vertex graph G = ( V , E ) there exists a partition V 1 ❴ ∪ . . . ❴ ∪ V t = V satisfying ✭ i ✮ t 0 ≤ t ≤ T 0 , ✭ ii ✮ | V i | = n / t, and ✭ iii ✮ for every U ⊆ V we have � � � � � ≤ ε n 2 . � � e ( U ) − d ( V i , V j ) | U ∩ V i || U ∩ V j | � � � � 1 ≤ i < j ≤ t � � Remarks T 0 = t 0 2 O ( 1 /ε 2 ) suffices Mathias Schacht (HU-Berlin) Graphs and limits November 2008