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¨

• dl 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¨

• dl et al., Hungarians, and Alon et al.

few years ago “different” hypergraph extensions by R¨

• dl 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¨

• dl et al., Hungarians, and Alon et al.

few years ago “different” hypergraph extensions by R¨

• dl 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¨

• dl et al., Hungarians, and Alon et al.

few years ago “different” hypergraph extensions by R¨

• dl 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¨

• dl et al., Hungarians, and Alon et al.

few years ago “different” hypergraph extensions by R¨

• dl 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¨

• dl et al., Hungarians, and Alon et al.

few years ago “different” hypergraph extensions by R¨

• dl 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¨

• dl et al., Hungarians, and Alon et al.

few years ago “different” hypergraph extensions by R¨

• dl 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 T0 such that for every n-vertex graph G = (V , E) there exists a partition V1 ❴ ∪ . . . ❴ ∪Vt = V satisfying ✭i ✮ t ≤ T0, ✭ ✮ ✭ ✮

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

A simple regularity lemma

Theorem (Frieze-Kannan ’99) For every ε > 0 exists T0 such that for every n-vertex graph G = (V , E) there exists a partition V1 ❴ ∪ . . . ❴ ∪Vt = V satisfying ✭i ✮ t ≤ T0, ✭ii ✮ |V1| ≤ · · · ≤ |Vt| ≤ |V1| + 1, ✭ ✮

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

A simple regularity lemma

Theorem (Frieze-Kannan ’99) For every ε > 0 exists T0 such that for every n-vertex graph G = (V , E) there exists a partition V1 ❴ ∪ . . . ❴ ∪Vt = V satisfying ✭i ✮ t ≤ T0, ✭ii ✮ |V1| ≤ · · · ≤ |Vt| ≤ |V1| + 1, and ✭iii ✮ for every U ⊆ V we have

• e(U) −
• 1≤i<j≤t

d(Vi, Vj)|U ∩ Vi||U ∩ Vj|

• ≤ εn2 .

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 = V1 ❴ ∪ . . . ❴ ∪Vt be a partition of V . We define the index of V by ✐♥❞(V) = 1 n2

• i<j

d2(Vi, Vj)|Vi||Vj| . ✭ ✮ ✭ ✮ ✭ ✮ ✐♥❞ ✐♥❞ ❴ ❴ ❴ ❴ ✭ ✮ ✭ ✮

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 = V1 ❴ ∪ . . . ❴ ∪Vt be a partition of V . We define the index of V by ✐♥❞(V) = 1 n2

• i<j

d2(Vi, Vj)|Vi||Vj| . Proof. Iteratively define partitions V1, . . . 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 = V1 ❴ ∪ . . . ❴ ∪Vt be a partition of V . We define the index of V by ✐♥❞(V) = 1 n2

• i<j

d2(Vi, Vj)|Vi||Vj| . Proof. Iteratively define partitions V1, . . . such that ✭i ✮ and ✭ii ✮ hold. If U does not satisfy ✭iii ✮ for Vi, then ✐♥❞(W) ≥ ✐♥❞(Vi) + ε2 for Q = (V1 ∩ U) ❴ ∪(V1 \ U) ❴ ∪ . . . ❴ ∪ (Vti ∩ U) ❴ ∪(Vti \ 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 = V1 ❴ ∪ . . . ❴ ∪Vt be a partition of V . We define the index of V by ✐♥❞(V) = 1 n2

• i<j

d2(Vi, Vj)|Vi||Vj| . Proof. Iteratively define partitions V1, . . . such that ✭i ✮ and ✭ii ✮ hold. If U does not satisfy ✭iii ✮ for Vi, then ✐♥❞(W) ≥ ✐♥❞(Vi) + ε2 for Q = (V1 ∩ U) ❴ ∪(V1 \ U) ❴ ∪ . . . ❴ ∪ (Vti ∩ U) ❴ ∪(Vti \ U) . “Massage” W and obtain Vi+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 = V1 ❴ ∪ . . . ❴ ∪Vt be a partition of V . We define the index of V by ✐♥❞(V) = 1 n2

• i<j

d2(Vi, Vj)|Vi||Vj| . Proof. Iteratively define partitions V1, . . . such that ✭i ✮ and ✭ii ✮ hold. If U does not satisfy ✭iii ✮ for Vi, then ✐♥❞(W) ≥ ✐♥❞(Vi) + ε2 for Q = (V1 ∩ U) ❴ ∪(V1 \ U) ❴ ∪ . . . ❴ ∪ (Vti ∩ U) ❴ ∪(Vti \ U) . “Massage” W and obtain Vi+1, which satisfies ✭i ✮, ✭ii ✮ and which “keeps” the index-increment. Procedure must end after at most O(1/ε2) iterations.

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

The FK-Lemma, revisited

Theorem For every ε > 0, t0 ∈ N exists T0 and n0 such that for every n-vertex graph G = (V , E) there exists a partition V1 ❴ ∪ . . . ❴ ∪Vt = V satisfying ✭i ✮ t0 ≤ t ≤ T0, ✭ii ✮ |Vi| = n/t, and ✭iii ✮ for every U ⊆ V we have

• e(U) −
• 1≤i<j≤t

d(Vi, Vj)|U ∩ Vi||U ∩ Vj|

• ≤ εn2 .

❴ ❴ ❴ ❴

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

The FK-Lemma, revisited

Theorem For every ε > 0, t0 ∈ N exists T0 and n0 such that for every n-vertex graph G = (V , E) there exists a partition V1 ❴ ∪ . . . ❴ ∪Vt = V satisfying ✭i ✮ t0 ≤ t ≤ T0, ✭ii ✮ |Vi| = n/t, and ✭iii ✮ for every U ⊆ V we have

• e(U) −
• 1≤i<j≤t

d(Vi, Vj)|U ∩ Vi||U ∩ Vj|

• ≤ εn2 .

Remarks T0 = t02O(1/ε2) suffices ❴ ❴ ❴ ❴

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

The FK-Lemma, revisited

Theorem For every ε > 0, t0 ∈ N exists T0 and n0 such that for every n-vertex graph G = (V , E) there exists a partition V1 ❴ ∪ . . . ❴ ∪Vt = V satisfying ✭i ✮ t0 ≤ t ≤ T0, ✭ii ✮ |Vi| = n/t, and ✭iii ✮ for every U ⊆ V we have

• e(U) −
• 1≤i<j≤t

d(Vi, Vj)|U ∩ Vi||U ∩ Vj|

• ≤ εn2 .

Remarks T0 = t02O(1/ε2) suffices the V1 ❴ ∪ . . . ❴ ∪Vt may refine any given equitable partition U1 ❴ ∪ . . . ❴ ∪Ut0

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Tao’s proof of Szemer´ edi’s lemma

Theorem (Tao’06) For every ε > 0 and δ: N → (0, 1] there exist T0 and n0 s.t. for every n-vertex graph G = (V , E) there exist partition U = (Ui)i∈[s] and refinement W = (Wi,j)i∈[s],j∈[t] s.t. ✭ ✮ ✭ ✮ ✭ ✮ ✐♥❞ ✐♥❞ ✭ ✮ ♣♦❧② ❚❖❲❊❘ ♣♦❧②

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Tao’s proof of Szemer´ edi’s lemma

Theorem (Tao’06) For every ε > 0 and δ: N → (0, 1] there exist T0 and n0 s.t. for every n-vertex graph G = (V , E) there exist partition U = (Ui)i∈[s] and refinement W = (Wi,j)i∈[s],j∈[t] s.t. ✭a ✮ U satisfies properties of the FK-Lemma for ε and T0, ✭b ✮ W satisfies properties of the FK-Lemma for δ(s) and T0, ✭ ✮ ✐♥❞ ✐♥❞ ✭ ✮ ♣♦❧② ❚❖❲❊❘ ♣♦❧②

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Tao’s proof of Szemer´ edi’s lemma

Theorem (Tao’06) For every ε > 0 and δ: N → (0, 1] there exist T0 and n0 s.t. for every n-vertex graph G = (V , E) there exist partition U = (Ui)i∈[s] and refinement W = (Wi,j)i∈[s],j∈[t] s.t. ✭a ✮ U satisfies properties of the FK-Lemma for ε and T0, ✭b ✮ W satisfies properties of the FK-Lemma for δ(s) and T0, ✭c ✮ ✐♥❞(W) ≤ ✐♥❞(U) + ε. ✭ ✮ ♣♦❧② ❚❖❲❊❘ ♣♦❧②

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Tao’s proof of Szemer´ edi’s lemma

Theorem (Tao’06) For every ε > 0 and δ: N → (0, 1] there exist T0 and n0 s.t. for every n-vertex graph G = (V , E) there exist partition U = (Ui)i∈[s] and refinement W = (Wi,j)i∈[s],j∈[t] s.t. ✭a ✮ U satisfies properties of the FK-Lemma for ε and T0, ✭b ✮ W satisfies properties of the FK-Lemma for δ(s) and T0, ✭c ✮ ✐♥❞(W) ≤ ✐♥❞(U) + ε. Proof.

1

apply FK-Lemma with ε and obtain U and s ✭ ✮ ♣♦❧② ❚❖❲❊❘ ♣♦❧②

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Tao’s proof of Szemer´ edi’s lemma

Theorem (Tao’06) For every ε > 0 and δ: N → (0, 1] there exist T0 and n0 s.t. for every n-vertex graph G = (V , E) there exist partition U = (Ui)i∈[s] and refinement W = (Wi,j)i∈[s],j∈[t] s.t. ✭a ✮ U satisfies properties of the FK-Lemma for ε and T0, ✭b ✮ W satisfies properties of the FK-Lemma for δ(s) and T0, ✭c ✮ ✐♥❞(W) ≤ ✐♥❞(U) + ε. Proof.

1

apply FK-Lemma with ε and obtain U and s

2

apply FK-Lemma with δ(s) and obtain refinement W ✭ ✮ ♣♦❧② ❚❖❲❊❘ ♣♦❧②

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Tao’s proof of Szemer´ edi’s lemma

Theorem (Tao’06) For every ε > 0 and δ: N → (0, 1] there exist T0 and n0 s.t. for every n-vertex graph G = (V , E) there exist partition U = (Ui)i∈[s] and refinement W = (Wi,j)i∈[s],j∈[t] s.t. ✭a ✮ U satisfies properties of the FK-Lemma for ε and T0, ✭b ✮ W satisfies properties of the FK-Lemma for δ(s) and T0, ✭c ✮ ✐♥❞(W) ≤ ✐♥❞(U) + ε. Proof.

1

apply FK-Lemma with ε and obtain U and s

2

apply FK-Lemma with δ(s) and obtain refinement W

3

if ✭c ✮ holds, then stop; otherwise replace U by W and goto Step 2 ♣♦❧② ❚❖❲❊❘ ♣♦❧②

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Tao’s proof of Szemer´ edi’s lemma

Theorem (Tao’06) For every ε > 0 and δ: N → (0, 1] there exist T0 and n0 s.t. for every n-vertex graph G = (V , E) there exist partition U = (Ui)i∈[s] and refinement W = (Wi,j)i∈[s],j∈[t] s.t. ✭a ✮ U satisfies properties of the FK-Lemma for ε and T0, ✭b ✮ W satisfies properties of the FK-Lemma for δ(s) and T0, ✭c ✮ ✐♥❞(W) ≤ ✐♥❞(U) + ε. Proof.

1

apply FK-Lemma with ε and obtain U and s

2

apply FK-Lemma with δ(s) and obtain refinement W

3

if ✭c ✮ holds, then stop; otherwise replace U by W and goto Step 2 Remarks bound on T0 depends of δ(·) ♣♦❧② ❚❖❲❊❘ ♣♦❧②

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Tao’s proof of Szemer´ edi’s lemma

Theorem (Tao’06) For every ε > 0 and δ: N → (0, 1] there exist T0 and n0 s.t. for every n-vertex graph G = (V , E) there exist partition U = (Ui)i∈[s] and refinement W = (Wi,j)i∈[s],j∈[t] s.t. ✭a ✮ U satisfies properties of the FK-Lemma for ε and T0, ✭b ✮ W satisfies properties of the FK-Lemma for δ(s) and T0, ✭c ✮ ✐♥❞(W) ≤ ✐♥❞(U) + ε. Proof.

1

apply FK-Lemma with ε and obtain U and s

2

apply FK-Lemma with δ(s) and obtain refinement W

3

if ✭c ✮ holds, then stop; otherwise replace U by W and goto Step 2 Remarks bound on T0 depends of δ(·) if δ(s) = ♣♦❧②(1/s), then T0 = ❚❖❲❊❘(♣♦❧②(1/ε)) suffices

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Szemer´ edi’s regularity lemma

Theorem (Szemer´ edi ’78) For every ε > 0 and t0 there exist T0 and n0 such that for every n-vertex graph G = (V , E) there exists a partition V1 ❴ ∪ . . . ❴ ∪Vt = V such that ✭i ✮ t0 ≤ t ≤ T0, ✭ii ✮ Vi = n/t, ✭ ✮ ❚❖❲❊❘ ❚❖❲❊❘

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Szemer´ edi’s regularity lemma

Theorem (Szemer´ edi ’78) For every ε > 0 and t0 there exist T0 and n0 such that for every n-vertex graph G = (V , E) there exists a partition V1 ❴ ∪ . . . ❴ ∪Vt = V such that ✭i ✮ t0 ≤ t ≤ T0, ✭ii ✮ Vi = n/t, and ✭iii ✮ all but at most εt2 pairs (Vi, Vj) are ε-regular, i.e., for all Ui ⊆ Vi and Uj ⊆ Vj we have |e(Ui, Uj) − d(Vi, Vj)|Ui||Uj|| ≤ ε “n t ”2 . ❚❖❲❊❘ ❚❖❲❊❘

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Szemer´ edi’s regularity lemma

Theorem (Szemer´ edi ’78) For every ε > 0 and t0 there exist T0 and n0 such that for every n-vertex graph G = (V , E) there exists a partition V1 ❴ ∪ . . . ❴ ∪Vt = V such that ✭i ✮ t0 ≤ t ≤ T0, ✭ii ✮ Vi = n/t, and ✭iii ✮ all but at most εt2 pairs (Vi, Vj) are ε-regular, i.e., for all Ui ⊆ Vi and Uj ⊆ Vj we have |e(Ui, Uj) − d(Vi, Vj)|Ui||Uj|| ≤ ε “n t ”2 . Remarks the “original” lemma ❚❖❲❊❘ ❚❖❲❊❘

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Szemer´ edi’s regularity lemma

Theorem (Szemer´ edi ’78) For every ε > 0 and t0 there exist T0 and n0 such that for every n-vertex graph G = (V , E) there exists a partition V1 ❴ ∪ . . . ❴ ∪Vt = V such that ✭i ✮ t0 ≤ t ≤ T0, ✭ii ✮ Vi = n/t, and ✭iii ✮ all but at most εt2 pairs (Vi, Vj) are ε-regular, i.e., for all Ui ⊆ Vi and Uj ⊆ Vj we have |e(Ui, Uj) − d(Vi, Vj)|Ui||Uj|| ≤ ε “n t ”2 . Remarks the “original” lemma Proof: apply Tao’s lemma with δ(s) ≪ ε/s2 and then U has the desired properties ❚❖❲❊❘ ❚❖❲❊❘

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Szemer´ edi’s regularity lemma

Theorem (Szemer´ edi ’78) For every ε > 0 and t0 there exist T0 and n0 such that for every n-vertex graph G = (V , E) there exists a partition V1 ❴ ∪ . . . ❴ ∪Vt = V such that ✭i ✮ t0 ≤ t ≤ T0, ✭ii ✮ Vi = n/t, and ✭iii ✮ all but at most εt2 pairs (Vi, Vj) are ε-regular, i.e., for all Ui ⊆ Vi and Uj ⊆ Vj we have |e(Ui, Uj) − d(Vi, Vj)|Ui||Uj|| ≤ ε “n t ”2 . Remarks the “original” lemma Proof: apply Tao’s lemma with δ(s) ≪ ε/s2 and then U has the desired properties in fact T0 = ❚❖❲❊❘(1/ε5) suffices ❚❖❲❊❘

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Szemer´ edi’s regularity lemma

Theorem (Szemer´ edi ’78) For every ε > 0 and t0 there exist T0 and n0 such that for every n-vertex graph G = (V , E) there exists a partition V1 ❴ ∪ . . . ❴ ∪Vt = V such that ✭i ✮ t0 ≤ t ≤ T0, ✭ii ✮ Vi = n/t, and ✭iii ✮ all but at most εt2 pairs (Vi, Vj) are ε-regular, i.e., for all Ui ⊆ Vi and Uj ⊆ Vj we have |e(Ui, Uj) − d(Vi, Vj)|Ui||Uj|| ≤ ε “n t ”2 . Remarks the “original” lemma Proof: apply Tao’s lemma with δ(s) ≪ ε/s2 and then U has the desired properties in fact T0 = ❚❖❲❊❘(1/ε5) suffices Gowers showed T0 = ❚❖❲❊❘(1/δc) for c = 1/16 is required

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Szemer´ edi’s regularity lemma

Theorem (Szemer´ edi ’78) For every ε > 0 and t0 there exist T0 and n0 such that for every n-vertex graph G = (V , E) there exists a partition V1 ❴ ∪ . . . ❴ ∪Vt = V such that ✭i ✮ t0 ≤ t ≤ T0, ✭ii ✮ Vi = n/t, and ✭iii ✮ all but at most εt2 pairs (Vi, Vj) are ε-regular, i.e., for all Ui ⊆ Vi and Uj ⊆ Vj we have |e(Ui, Uj) − d(Vi, Vj)|Ui||Uj|| ≤ ε “n t ”2 . Remarks the “original” lemma Proof: apply Tao’s lemma with δ(s) ≪ ε/s2 and then U has the desired properties in fact T0 = ❚❖❲❊❘(1/ε5) suffices Gowers showed T0 = ❚❖❲❊❘(1/δc) for c = 1/16 is required ⇒ ε ≫ 1/T0 is unavoidable

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Let’s iterate some more

Theorem (Alon, Fischer, Krivelevich, Szegedy ’00) For every ε > 0 and δ: N → (0, 1] there exist T0 and n0 such that for every n-vertex graph G = (V , E) there exist partition U = (Ui)i∈[s] and refinement W = (Wi,j)i∈[s],j∈[t] such that ✭a ✮ U satisfies properties of the Sz-Lemma for ε and T0, ✭b ✮ W satisfies properties of the Sz-Lemma for δ(s) and T0, ✭c ✮ ✐♥❞(W) ≤ ✐♥❞(U) + ε. ✭ ✮ ❲❖❲ ♣♦❧② ♣♦❧②

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Let’s iterate some more

Theorem (Alon, Fischer, Krivelevich, Szegedy ’00) For every ε > 0 and δ: N → (0, 1] there exist T0 and n0 such that for every n-vertex graph G = (V , E) there exist partition U = (Ui)i∈[s] and refinement W = (Wi,j)i∈[s],j∈[t] such that ✭a ✮ U satisfies properties of the Sz-Lemma for ε and T0, ✭b ✮ W satisfies properties of the Sz-Lemma for δ(s) and T0, ✭c ✮ ✐♥❞(W) ≤ ✐♥❞(U) + ε. Remarks Proof: iterate Sz-Lemma until ✭c ✮ holds ❲❖❲ ♣♦❧② ♣♦❧②

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Let’s iterate some more

Theorem (Alon, Fischer, Krivelevich, Szegedy ’00) For every ε > 0 and δ: N → (0, 1] there exist T0 and n0 such that for every n-vertex graph G = (V , E) there exist partition U = (Ui)i∈[s] and refinement W = (Wi,j)i∈[s],j∈[t] such that ✭a ✮ U satisfies properties of the Sz-Lemma for ε and T0, ✭b ✮ W satisfies properties of the Sz-Lemma for δ(s) and T0, ✭c ✮ ✐♥❞(W) ≤ ✐♥❞(U) + ε. Remarks Proof: iterate Sz-Lemma until ✭c ✮ holds bound on T0 deteriorates to ❲❖❲(♣♦❧②(1/ε)) for δ(s) = ♣♦❧②(1/s).

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Reduced graph/Clustergraph

Definition (Reduced Graph) Let G = (V , E) be a graph, V = (Vi)i∈[t] be a partition of V , and ε > 0 We define the reduced graph R = R(V, ε) as a weighted graph with vertex set [t] and edge weights w : [t]

2

• → {0, ε, 2ε, . . . , ⌊1/ε⌋ε} such that

d(Vi, Vj) − ε < w(i, j) ≤ d(Vi, Vj) for all 1 ≤ i < j ≤ t.

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Reduced graph/Clustergraph

Definition (Reduced Graph) Let G = (V , E) be a graph, V = (Vi)i∈[t] be a partition of V , and ε > 0 We define the reduced graph R = R(V, ε) as a weighted graph with vertex set [t] and edge weights w : [t]

2

• → {0, ε, 2ε, . . . , ⌊1/ε⌋ε} such that

d(Vi, Vj) − ε < w(i, j) ≤ d(Vi, Vj) for all 1 ≤ i < j ≤ t. Idea If R(V, ε) is the reduced graph of some “regular partition” of G, then certain graph parameters/properties of G can be approximated by studying R(V, ε).

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Reduced graph/Clustergraph

Definition (Reduced Graph) Let G = (V , E) be a graph, V = (Vi)i∈[t] be a partition of V , and ε > 0 We define the reduced graph R = R(V, ε) as a weighted graph with vertex set [t] and edge weights w : [t]

2

• → {0, ε, 2ε, . . . , ⌊1/ε⌋ε} such that

d(Vi, Vj) − ε < w(i, j) ≤ d(Vi, Vj) for all 1 ≤ i < j ≤ t. Idea If R(V, ε) is the reduced graph of some “regular partition” of G, then certain graph parameters/properties of G can be approximated by studying R(V, ε). MAX-CUT can be approximated up to an additive error of εn2 by studying the weighted MAX-CUT of R(V, ε) of an FK-partition

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Reduced graph/Clustergraph

Definition (Reduced Graph) Let G = (V , E) be a graph, V = (Vi)i∈[t] be a partition of V , and ε > 0 We define the reduced graph R = R(V, ε) as a weighted graph with vertex set [t] and edge weights w : [t]

2

• → {0, ε, 2ε, . . . , ⌊1/ε⌋ε} such that

d(Vi, Vj) − ε < w(i, j) ≤ d(Vi, Vj) for all 1 ≤ i < j ≤ t. Idea If R(V, ε) is the reduced graph of some “regular partition” of G, then certain graph parameters/properties of G can be approximated by studying R(V, ε). MAX-CUT can be approximated up to an additive error of εn2 by studying the weighted MAX-CUT of R(V, ε) of an FK-partition if R(V, ε) restricted to edges of positive weight of an Sz-partition is not k-colorable, then G is not k-colorable

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Subgraph frequencies/global counting lemma

Lemma (Subgraph frequencies) For every γ > 0 and every graph F with vertex set [k], there exists ε > 0 and n0, such that if V is a FK-/Sz-partition of G with t parts and reduced graph R(V, ε), then ★{F ⊆ G} =

• x1,...,xk
• ij∈E(F)

w(xi, xj) × “n t ”k ± γnk = ★{F ⊆ R} × “n t ”k ± γnk . ★ ★

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Subgraph frequencies/global counting lemma

Lemma (Subgraph frequencies) For every γ > 0 and every graph F with vertex set [k], there exists ε > 0 and n0, such that if V is a FK-/Sz-partition of G with t parts and reduced graph R(V, ε), then ★{F ⊆ G} =

• x1,...,xk
• ij∈E(F)

w(xi, xj) × “n t ”k ± γnk = ★{F ⊆ R} × “n t ”k ± γnk . Lemma (Subgraph frequencies - induced) For every γ > 0 and every graph F with vertex set [k], there exists ε > 0 and n0, such that if V is a FK-/Sz-partition of G with t parts and reduced graph R(V, ε), then ★{F≤G} =

• x1,...,xk
• ij∈E(F)

w(xi, xj)

• ij∈E(F)

(1 − w(xi, xj)) × “n t ”k ± γnk . = ★{F≤R} × “n t ”k ± γnk .

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

(local) Counting Lemma

Lemma (Counting lemma) For every γ > 0 and every graph F with vertex set [k], there exists ε > 0 and n0, such that if V = (Vi)i∈[t] is a Sz-partition of G, and x1, . . . , xk ∈ [t] such that (Vxi, Vxj) is ε-regular whenever ij ∈ E(F) then ★{F ⊆ G[Vx1, . . . , Vxk]} =

• ij∈E(F)

w(xi, xj) × n t k ± γnk . ★

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

(local) Counting Lemma

Lemma (Counting lemma) For every γ > 0 and every graph F with vertex set [k], there exists ε > 0 and n0, such that if V = (Vi)i∈[t] is a Sz-partition of G, and x1, . . . , xk ∈ [t] such that (Vxi, Vxj) is ε-regular whenever ij ∈ E(F) then ★{F ⊆ G[Vx1, . . . , Vxk]} =

• ij∈E(F)

w(xi, xj) × n t k ± γnk . Lemma (Counting lemma - induced) For every γ > 0 and every graph F with vertex set [k], there exists ε > 0 and n0, such that if V = (Vi)i∈[t] is a Sz-partition of G, and x1, . . . , xk ∈ [t] such that (Vxi, Vxj) is ε-regular for all ij ∈ [k]

2

• then

★{F ≤ G[Vx1, . . . , Vxk]} =

• ij∈E(F)

w(xi, xj)

• ij∈E(F)

(1 − w(xi, xj)) × n t k ± γnk .

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Removal lemma

Theorem (Removal lemma) For every η > 0, every k and every graph F with k vertices there exists c > 0 and n0 such that for every graph G = (V , E) with n ≥ n0 vertices and ★{F ⊆ G} ≤ cnk there exists a F-free subgraph G ′ = (V , E ′) such that |E \ E ′| ≤ ηn2. Remarks easy consequence of Szemer´ edi’s regularity lemma and counting lemma

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Removal lemma

Theorem (Removal lemma) For every η > 0, every k and every graph F with k vertices there exists c > 0 and n0 such that for every graph G = (V , E) with n ≥ n0 vertices and ★{F ⊆ G} ≤ cnk there exists a F-free subgraph G ′ = (V , E ′) such that |E \ E ′| ≤ ηn2. Remarks easy consequence of Szemer´ edi’s regularity lemma and counting lemma first proved for triangles by Ruzsa and Szemer´ edi ’78 generalized to arbitrary F by Erd˝

• s, Frankl, and R¨
• dl ’86

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Removal lemma

Theorem (Removal lemma) For every η > 0, every k and every graph F with k vertices there exists c > 0 and n0 such that for every graph G = (V , E) with n ≥ n0 vertices and ★{F ⊆ G} ≤ cnk there exists a F-free subgraph G ′ = (V , E ′) such that |E \ E ′| ≤ ηn2. Remarks easy consequence of Szemer´ edi’s regularity lemma and counting lemma first proved for triangles by Ruzsa and Szemer´ edi ’78 generalized to arbitrary F by Erd˝

• s, Frankl, and R¨
• dl ’86

hypergraph generalizations imply Szemer´ edi’s theorem on arithmetic progressions and its multidimensional version due to F¨ urstenberg and Katznelson (Wednesday)

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Removal lemma

Theorem (Removal lemma) For every η > 0, every k and every graph F with k vertices there exists c > 0 and n0 such that for every graph G = (V , E) with n ≥ n0 vertices and ★{F ⊆ G} ≤ cnk there exists a F-free subgraph G ′ = (V , E ′) such that |E \ E ′| ≤ ηn2. Remarks easy consequence of Szemer´ edi’s regularity lemma and counting lemma first proved for triangles by Ruzsa and Szemer´ edi ’78 generalized to arbitrary F by Erd˝

• s, Frankl, and R¨
• dl ’86

hypergraph generalizations imply Szemer´ edi’s theorem on arithmetic progressions and its multidimensional version due to F¨ urstenberg and Katznelson (Wednesday) Later: we discuss infinite, induced generalizations for graphs

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Where are we?

FK-Lemma e(U) behaves as expected by R(V, ε) if |U| ≫ εn ★{F ⊆ G} can be read from R(V, ε) (global counting lemma) number of classes is exponential in ♣♦❧②(1/ε) ★ ♣♦❧② ♣♦❧②

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Where are we?

FK-Lemma e(U) behaves as expected by R(V, ε) if |U| ≫ εn ★{F ⊆ G} can be read from R(V, ε) (global counting lemma) number of classes is exponential in ♣♦❧②(1/ε) Sz-Lemma . = iterated FK-Lemma e(U) behaves as expected by R(V, ε) if |U| ≫ εn/t ★{F ⊆ G[Vxi , . . . , Vxk ] can be read from R(V, ε) (local counting lemma) number of classes is a tower of height ♣♦❧②(1/ε) ♣♦❧②

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Where are we?

FK-Lemma e(U) behaves as expected by R(V, ε) if |U| ≫ εn ★{F ⊆ G} can be read from R(V, ε) (global counting lemma) number of classes is exponential in ♣♦❧②(1/ε) Sz-Lemma . = iterated FK-Lemma e(U) behaves as expected by R(V, ε) if |U| ≫ εn/t ★{F ⊆ G[Vxi , . . . , Vxk ] can be read from R(V, ε) (local counting lemma) number of classes is a tower of height ♣♦❧②(1/ε) AFKS-Lemma . = iterated Sz-Lemma random selection of representatives (Wi,ji )i∈[s] from the finer partition satisfies: all pairs are δ(s)-regular and most pairs satisfy |d(Wi,ji , Wi ′,ji ′ ) − d(Vi, Vi ′)| ≤ ε ⇒ counting lemma for graphs F with f (s) vertices is applicable ♣♦❧②

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Where are we?

FK-Lemma e(U) behaves as expected by R(V, ε) if |U| ≫ εn ★{F ⊆ G} can be read from R(V, ε) (global counting lemma) number of classes is exponential in ♣♦❧②(1/ε) Sz-Lemma . = iterated FK-Lemma e(U) behaves as expected by R(V, ε) if |U| ≫ εn/t ★{F ⊆ G[Vxi , . . . , Vxk ] can be read from R(V, ε) (local counting lemma) number of classes is a tower of height ♣♦❧②(1/ε) AFKS-Lemma . = iterated Sz-Lemma random selection of representatives (Wi,ji )i∈[s] from the finer partition satisfies: all pairs are δ(s)-regular and most pairs satisfy |d(Wi,ji , Wi ′,ji ′ ) − d(Vi, Vi ′)| ≤ ε ⇒ counting lemma for graphs F with f (s) vertices is applicable number of classes is a ♣♦❧②(1/ε) iterated tower

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Where are we?

FK-Lemma e(U) behaves as expected by R(V, ε) if |U| ≫ εn ★{F ⊆ G} can be read from R(V, ε) (global counting lemma) number of classes is exponential in ♣♦❧②(1/ε) Sz-Lemma . = iterated FK-Lemma e(U) behaves as expected by R(V, ε) if |U| ≫ εn/t ★{F ⊆ G[Vxi , . . . , Vxk ] can be read from R(V, ε) (local counting lemma) number of classes is a tower of height ♣♦❧②(1/ε) AFKS-Lemma . = iterated Sz-Lemma random selection of representatives (Wi,ji )i∈[s] from the finer partition satisfies: all pairs are δ(s)-regular and most pairs satisfy |d(Wi,ji , Wi ′,ji ′ ) − d(Vi, Vi ′)| ≤ ε ⇒ counting lemma for graphs F with f (s) vertices is applicable number of classes is a ♣♦❧②(1/ε) iterated tower (or worse)

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Some related lemmas

Szemer´ edi’s original lemma from 1975 stated for bipartite graphs G = (A ❴ ∪B, E) only there exists a partition of A = (Ai)i∈[s] of A such that for every Ai there exists a partition Bi = (Bij)j∈[t] of B such that . . . t ≤ T0 = 2O(♣♦❧②(1/ε) suffices

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Some related lemmas

Szemer´ edi’s original lemma from 1975 stated for bipartite graphs G = (A ❴ ∪B, E) only there exists a partition of A = (Ai)i∈[s] of A such that for every Ai there exists a partition Bi = (Bij)j∈[t] of B such that . . . t ≤ T0 = 2O(♣♦❧②(1/ε) suffices Duke, Lefmann, R¨

• dl ’95

multipartite, “cleaner” version of Szemer´ edi’s original lemma Corollary: Subgraph frequencies can be approximated efficiently for graphs F up to Ω(√log log n) vertices

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Some related lemmas

Szemer´ edi’s original lemma from 1975 stated for bipartite graphs G = (A ❴ ∪B, E) only there exists a partition of A = (Ai)i∈[s] of A such that for every Ai there exists a partition Bi = (Bij)j∈[t] of B such that . . . t ≤ T0 = 2O(♣♦❧②(1/ε) suffices Duke, Lefmann, R¨

• dl ’95

multipartite, “cleaner” version of Szemer´ edi’s original lemma Corollary: Subgraph frequencies can be approximated efficiently for graphs F up to Ω(√log log n) vertices R¨

• dl et al.

stronger notion of ε-regularity developed for work on hypergraphs (U, V ) is (ε, r)-regular if for all U1, . . . , Ur ⊆ U and V1, . . . , Vr ⊆ V ˛ ˛ ˛ ˛ ˛ ˛ ˛ [

i∈[r]

E(Ui, Vi) ˛ ˛ ˛ − d(U, V ) ˛ ˛ ˛ [

i∈[r]

Ui × Vi ˛ ˛ ˛ ˛ ˛ ˛ ˛ ≤ ε|U||V | Thm: ∀ε > 0, r : N → N ∃T0 s.t.

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Some related lemmas

Szemer´ edi’s original lemma from 1975 stated for bipartite graphs G = (A ❴ ∪B, E) only there exists a partition of A = (Ai)i∈[s] of A such that for every Ai there exists a partition Bi = (Bij)j∈[t] of B such that . . . t ≤ T0 = 2O(♣♦❧②(1/ε) suffices Duke, Lefmann, R¨

• dl ’95

multipartite, “cleaner” version of Szemer´ edi’s original lemma Corollary: Subgraph frequencies can be approximated efficiently for graphs F up to Ω(√log log n) vertices R¨

• dl et al.

stronger notion of ε-regularity developed for work on hypergraphs (U, V ) is (ε, r)-regular if for all U1, . . . , Ur ⊆ U and V1, . . . , Vr ⊆ V ˛ ˛ ˛ ˛ ˛ ˛ ˛ [

i∈[r]

E(Ui, Vi) ˛ ˛ ˛ − d(U, V ) ˛ ˛ ˛ [

i∈[r]

Ui × Vi ˛ ˛ ˛ ˛ ˛ ˛ ˛ ≤ ε|U||V | Thm: ∀ε > 0, r : N → N ∃T0 s.t. . . . all but εt2 pairs are (ε, r(t))-regular

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Some related lemmas

Szemer´ edi’s original lemma from 1975 stated for bipartite graphs G = (A ❴ ∪B, E) only there exists a partition of A = (Ai)i∈[s] of A such that for every Ai there exists a partition Bi = (Bij)j∈[t] of B such that . . . t ≤ T0 = 2O(♣♦❧②(1/ε) suffices Duke, Lefmann, R¨

• dl ’95

multipartite, “cleaner” version of Szemer´ edi’s original lemma Corollary: Subgraph frequencies can be approximated efficiently for graphs F up to Ω(√log log n) vertices R¨

• dl et al.

stronger notion of ε-regularity developed for work on hypergraphs (U, V ) is (ε, r)-regular if for all U1, . . . , Ur ⊆ U and V1, . . . , Vr ⊆ V ˛ ˛ ˛ ˛ ˛ ˛ ˛ [

i∈[r]

E(Ui, Vi) ˛ ˛ ˛ − d(U, V ) ˛ ˛ ˛ [

i∈[r]

Ui × Vi ˛ ˛ ˛ ˛ ˛ ˛ ˛ ≤ ε|U||V | Thm: ∀ε > 0, r : N → N ∃T0 s.t. . . . all but εt2 pairs are (ε, r(t))-regular equivalent to the AFKS-Lemma

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Regular approximation lemma

Theorem (R¨

• dl & S. ’07)

For every ν > 0 and ε: N → (0, 1] there exists T0 and n0 such that for every graph G = (V , E) on n ≥ n0 vertices there exists a partition V = (Vi)i∈[t] and a graph H = (V , E ′) such that V is a Sz-partition for H for ε(t)

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Regular approximation lemma

Theorem (R¨

• dl & S. ’07)

For every ν > 0 and ε: N → (0, 1] there exists T0 and n0 such that for every graph G = (V , E) on n ≥ n0 vertices there exists a partition V = (Vi)i∈[t] and a graph H = (V , E ′) such that V is a Sz-partition for H for ε(t) and |E(H)△E(G)| ≤ νn2.

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Regular approximation lemma

Theorem (R¨

• dl & S. ’07)

For every ν > 0 and ε: N → (0, 1] there exists T0 and n0 such that for every graph G = (V , E) on n ≥ n0 vertices there exists a partition V = (Vi)i∈[t] and a graph H = (V , E ′) such that V is a Sz-partition for H for ε(t) and |E(H)△E(G)| ≤ νn2. Remarks Consequence of the AFKS- or (ε, r)-Lemma

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Regular approximation lemma

Theorem (R¨

• dl & S. ’07)

For every ν > 0 and ε: N → (0, 1] there exists T0 and n0 such that for every graph G = (V , E) on n ≥ n0 vertices there exists a partition V = (Vi)i∈[t] and a graph H = (V , E ′) such that V is a Sz-partition for H for ε(t) and |E(H)△E(G)| ≤ νn2. Remarks Consequence of the AFKS- or (ε, r)-Lemma here ε(t) ≪ 1/t is allowed, despite Gowers’ lower bound

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Regular approximation lemma

Theorem (R¨

• dl & S. ’07)

For every ν > 0 and ε: N → (0, 1] there exists T0 and n0 such that for every graph G = (V , E) on n ≥ n0 vertices there exists a partition V = (Vi)i∈[t] and a graph H = (V , E ′) such that V is a Sz-partition for H for ε(t) and |E(H)△E(G)| ≤ νn2. Remarks Consequence of the AFKS- or (ε, r)-Lemma here ε(t) ≪ 1/t is allowed, despite Gowers’ lower bound hypergraph extensions exist

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Regular approximation lemma

Theorem (R¨

• dl & S. ’07)

For every ν > 0 and ε: N → (0, 1] there exists T0 and n0 such that for every graph G = (V , E) on n ≥ n0 vertices there exists a partition V = (Vi)i∈[t] and a graph H = (V , E ′) such that V is a Sz-partition for H for ε(t) and |E(H)△E(G)| ≤ νn2. Remarks Consequence of the AFKS- or (ε, r)-Lemma here ε(t) ≪ 1/t is allowed, despite Gowers’ lower bound hypergraph extensions exist Question

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Regular approximation lemma

Theorem (R¨

• dl & S. ’07)

For every ν > 0 and ε: N → (0, 1] there exists T0 and n0 such that for every graph G = (V , E) on n ≥ n0 vertices there exists a partition V = (Vi)i∈[t] and a graph H = (V , E ′) such that V is a Sz-partition for H for ε(t) and |E(H)△E(G)| ≤ νn2. Remarks Consequence of the AFKS- or (ε, r)-Lemma here ε(t) ≪ 1/t is allowed, despite Gowers’ lower bound hypergraph extensions exist Question Shall we iterate the AFKS-/(δ, r)-/approximation-lemma?

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

How to regularize with ε = 0?

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

How to regularize with ε = 0?

Idea Iterate FK-/Sz-/AFKS-Lemma infinitely often

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

How to regularize with ε = 0?

Idea Iterate FK-/Sz-/AFKS-Lemma infinitely often But

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

How to regularize with ε = 0?

Idea Iterate FK-/Sz-/AFKS-Lemma infinitely often But what could that mean:

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

How to regularize with ε = 0?

Idea Iterate FK-/Sz-/AFKS-Lemma infinitely often But what could that mean:

With what ε do we regularize?

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

How to regularize with ε = 0?

Idea Iterate FK-/Sz-/AFKS-Lemma infinitely often But what could that mean:

With what ε do we regularize?

• seq. εℓ → 0

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

How to regularize with ε = 0?

Idea Iterate FK-/Sz-/AFKS-Lemma infinitely often But what could that mean:

With what ε do we regularize?

• seq. εℓ → 0

Do we want to talk about infinite graphs?

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

How to regularize with ε = 0?

Idea Iterate FK-/Sz-/AFKS-Lemma infinitely often But what could that mean:

With what ε do we regularize?

• seq. εℓ → 0

Do we want to talk about infinite graphs? NO Consider an infinite sequence (Gℓ)ℓ∈N of graphs instead

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

How to regularize with ε = 0?

Idea Iterate FK-/Sz-/AFKS-Lemma infinitely often But what could that mean:

With what ε do we regularize?

• seq. εℓ → 0

Do we want to talk about infinite graphs? NO Consider an infinite sequence (Gℓ)ℓ∈N of graphs instead

Regularize every graph of (Gℓ)ℓ∈N with ε1 ⇒ ε1-regular partitions Vℓ

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

How to regularize with ε = 0?

Idea Iterate FK-/Sz-/AFKS-Lemma infinitely often But what could that mean:

With what ε do we regularize?

• seq. εℓ → 0

Do we want to talk about infinite graphs? NO Consider an infinite sequence (Gℓ)ℓ∈N of graphs instead

Regularize every graph of (Gℓ)ℓ∈N with ε1 ⇒ ε1-regular partitions Vℓ Since there are only finitely many possible reduced graphs,

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

How to regularize with ε = 0?

Idea Iterate FK-/Sz-/AFKS-Lemma infinitely often But what could that mean:

With what ε do we regularize?

• seq. εℓ → 0

Do we want to talk about infinite graphs? NO Consider an infinite sequence (Gℓ)ℓ∈N of graphs instead

Regularize every graph of (Gℓ)ℓ∈N with ε1 ⇒ ε1-regular partitions Vℓ Since there are only finitely many possible reduced graphs, there must be a reduced graph R1 such that R(Vℓ, ε1) = R1 for infinitely many graphs Gℓ from the sequence

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

How to regularize with ε = 0?

Idea Iterate FK-/Sz-/AFKS-Lemma infinitely often But what could that mean:

With what ε do we regularize?

• seq. εℓ → 0

Do we want to talk about infinite graphs? NO Consider an infinite sequence (Gℓ)ℓ∈N of graphs instead

Regularize every graph of (Gℓ)ℓ∈N with ε1 ⇒ ε1-regular partitions Vℓ Since there are only finitely many possible reduced graphs, there must be a reduced graph R1 such that R(Vℓ, ε1) = R1 for infinitely many graphs Gℓ from the sequence pass to such an infinite subsequence and regularize with ε2 with initial partition given by the first regularization

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

How to regularize with ε = 0?

Idea Iterate FK-/Sz-/AFKS-Lemma infinitely often But what could that mean:

With what ε do we regularize?

• seq. εℓ → 0

Do we want to talk about infinite graphs? NO Consider an infinite sequence (Gℓ)ℓ∈N of graphs instead

Regularize every graph of (Gℓ)ℓ∈N with ε1 ⇒ ε1-regular partitions Vℓ Since there are only finitely many possible reduced graphs, there must be a reduced graph R1 such that R(Vℓ, ε1) = R1 for infinitely many graphs Gℓ from the sequence pass to such an infinite subsequence and regularize with ε2 with initial partition given by the first regularization

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

How to regularize with ε = 0?

Idea Iterate FK-/Sz-/AFKS-Lemma infinitely often But what could that mean:

With what ε do we regularize?

• seq. εℓ → 0

Do we want to talk about infinite graphs? NO Consider an infinite sequence (Gℓ)ℓ∈N of graphs instead

Regularize every graph of (Gℓ)ℓ∈N with ε1 ⇒ ε1-regular partitions Vℓ Since there are only finitely many possible reduced graphs, there must be a reduced graph R1 such that R(Vℓ, ε1) = R1 for infinitely many graphs Gℓ from the sequence pass to such an infinite subsequence and regularize with ε2 with initial partition given by the first regularization . . . What do we get in the end?

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

How to regularize with ε = 0?

Idea Iterate FK-/Sz-/AFKS-Lemma infinitely often But what could that mean:

With what ε do we regularize?

• seq. εℓ → 0

Do we want to talk about infinite graphs? NO Consider an infinite sequence (Gℓ)ℓ∈N of graphs instead

Regularize every graph of (Gℓ)ℓ∈N with ε1 ⇒ ε1-regular partitions Vℓ Since there are only finitely many possible reduced graphs, there must be a reduced graph R1 such that R(Vℓ, ε1) = R1 for infinitely many graphs Gℓ from the sequence pass to such an infinite subsequence and regularize with ε2 with initial partition given by the first regularization . . . What do we get in the end? sequences (Rℓ)ℓ∈N and (G ′

ℓ)ℓ∈N

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Limit object

reduced graphs Rℓ can be viewd as symmetric matrices ❞ ❞ ❞ ❞ ★ ❞

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Limit object

reduced graphs Rℓ can be viewd as symmetric matrices which naturally correspond to measurable, symmetric step-functions Rℓ : [0, 1]2 → [0, 1]: ❞ ❞ ❞ ❞ ★ ❞

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Limit object

reduced graphs Rℓ can be viewd as symmetric matrices which naturally correspond to measurable, symmetric step-functions Rℓ : [0, 1]2 → [0, 1]: split [0, 1] into tℓ equal intervals and set Rℓ

• (i − 1)/tℓ, i/tℓ] × (j − 1)/tℓ, j/tℓ]
• ≡ wRℓ(i, j) .

pointwise limit R(x, y) = limℓ→∞ Rℓ(x, y) exists almost everywhere, since i/tℓ

(i−1)/tℓ

j/tℓ

(j−1)/tℓ

Rℓ(x, y)❞x❞y = i/tℓ

(i−1)/tℓ

j/tℓ

(j−1)/tℓ

Rℓ+1(x, y)❞x❞y ★ ❞

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Limit object

reduced graphs Rℓ can be viewd as symmetric matrices which naturally correspond to measurable, symmetric step-functions Rℓ : [0, 1]2 → [0, 1]: split [0, 1] into tℓ equal intervals and set Rℓ

• (i − 1)/tℓ, i/tℓ] × (j − 1)/tℓ, j/tℓ]
• ≡ wRℓ(i, j) .

pointwise limit R(x, y) = limℓ→∞ Rℓ(x, y) exists almost everywhere, since i/tℓ

(i−1)/tℓ

j/tℓ

(j−1)/tℓ

Rℓ(x, y)❞x❞y = i/tℓ

(i−1)/tℓ

j/tℓ

(j−1)/tℓ

Rℓ+1(x, y)❞x❞y moreover for every fixed graph F with V (F) = [k] we have lim

ℓ→∞ t(F, G ′ ℓ) = lim ℓ→∞

★{F ⊆ G ′

ℓ}

nk = t(F, R) where t(F, R) =

• x1,...,xk
• ij∈E(F)

R(xi, xj)❞x .

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Summarizing

Theorem (Lov´ asz & Szegedy ’06) For every infinite sequence of graphs (Gℓ)ℓ∈N there exists an infinite subsequence (G ′

ℓ)ℓ∈N and a symmetric, measurable function

R : [0, 1]2 → [0, 1] such that ✭i ✮ t(F, R) = limℓ→∞ t(F, G ′

ℓ) for all graphs F

✭ ✮

✐♥❞ ✐♥❞

✭ ✮ ❞ ❞

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Summarizing

Theorem (Lov´ asz & Szegedy ’06) For every infinite sequence of graphs (Gℓ)ℓ∈N there exists an infinite subsequence (G ′

ℓ)ℓ∈N and a symmetric, measurable function

R : [0, 1]2 → [0, 1] such that ✭i ✮ t(F, R) = limℓ→∞ t(F, G ′

ℓ) for all graphs F

✭ii ✮ t✐♥❞(F, R) = limℓ→∞ t✐♥❞(F, G ′

ℓ) for all graphs F

✭ ✮ ❞ ❞

Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Summarizing

Theorem (Lov´ asz & Szegedy ’06) For every infinite sequence of graphs (Gℓ)ℓ∈N there exists an infinite subsequence (G ′

ℓ)ℓ∈N and a symmetric, measurable function

R : [0, 1]2 → [0, 1] such that ✭i ✮ t(F, R) = limℓ→∞ t(F, G ′

ℓ) for all graphs F

✭ii ✮ t✐♥❞(F, R) = limℓ→∞ t✐♥❞(F, G ′

ℓ) for all graphs F

✭iii ✮ for every U ⊆ [0, 1] we have

• U×U

R(x, y)❞x❞y = lim

ℓ→∞ dG ′

ℓ (U) . Mathias Schacht (HU-Berlin) Graphs and limits November 2008

Summarizing

Theorem (Lov´ asz & Szegedy ’06) For every infinite sequence of graphs (Gℓ)ℓ∈N there exists an infinite subsequence (G ′

ℓ)ℓ∈N and a symmetric, measurable function

R : [0, 1]2 → [0, 1] such that ✭i ✮ t(F, R) = limℓ→∞ t(F, G ′

ℓ) for all graphs F

✭ii ✮ t✐♥❞(F, R) = limℓ→∞ t✐♥❞(F, G ′

ℓ) for all graphs F

✭iii ✮ for every U ⊆ [0, 1] we have

• U×U

R(x, y)❞x❞y = lim

ℓ→∞ dG ′

ℓ (U) . Mathias Schacht (HU-Berlin) Graphs and limits November 2008