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

Property testing and hypergraph regularity lemmas

Mathias Schacht

Institut f¨ ur Informatik Humboldt-Universit¨ at zu Berlin

November 2008

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Outline

1 Property testing for graphs

Removal lemma and its generalizations Proof of generalized removal lemma

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Outline

1 Property testing for graphs

Removal lemma and its generalizations Proof of generalized removal lemma

2 Quasi-random hypergraphs

Quasi-random graphs Three possible extensions The “right” extension

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity 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.

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity 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

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity 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 Short version: η-far from F-free ⇒ “many” copies of F

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

The density version of the Gallai-Witt theorem

Removal lemma implies (based on a construction of Solymosi ’04):

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

The density version of the Gallai-Witt theorem

Removal lemma implies (based on a construction of Solymosi ’04): Theorem (F¨ urstenberg & Katznelson ’78) ∀d, ∀k, ∀δ ∃n0 such that, ∀n ≥ n0, if S ⊂ [n]d with |S| ≥ δnd then S contains k × k × · · · × k

• d times

regular sublattice.

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Density version of the Gallai-Witt for d = 2 and k = 3

If |S| ≥ δn2, then . . .

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Density version of the Gallai-Witt for d = 2 and k = 3

If |S| ≥ δn2, then . . .

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Density version of the Gallai-Witt for d = 2 and k = 3

If |S| ≥ δn2, then . . . S contains 3 × 3 regular sublattice.

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Generalized removal lemma

Theorem (Alon & Shapira ’05) For every η > 0 and every family of graphs F there exists c > 0, L, and n0 such that every n-vertex graph G = (V , E) that is η-far from being induced F-free, there exists a ℓ-vertex graph F ∈ F with ℓ ≤ L such that ★{F ≤ G} ≥ cnℓ . ❋♦r❜✐♥❞

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Generalized removal lemma

Theorem (Alon & Shapira ’05) For every η > 0 and every family of graphs F there exists c > 0, L, and n0 such that every n-vertex graph G = (V , E) that is η-far from being induced F-free, there exists a ℓ-vertex graph F ∈ F with ℓ ≤ L such that ★{F ≤ G} ≥ cnℓ . Remarks Short version: η-far from ind. F-free ⇒ “many” copies of a small F ∈ F ❋♦r❜✐♥❞

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Generalized removal lemma

Theorem (Alon & Shapira ’05) For every η > 0 and every family of graphs F there exists c > 0, L, and n0 such that every n-vertex graph G = (V , E) that is η-far from being induced F-free, there exists a ℓ-vertex graph F ∈ F with ℓ ≤ L such that ★{F ≤ G} ≥ cnℓ . Remarks Short version: η-far from ind. F-free ⇒ “many” copies of a small F ∈ F Alon, Fischer, Krivelevich, and M. Szegedy ’00 for single graph F alternative proof by Lov´ asz and B. Szegedy ’04 R¨

• dl and S. obtained hypergraph generalization

❋♦r❜✐♥❞

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Generalized removal lemma

Theorem (Alon & Shapira ’05) For every η > 0 and every family of graphs F there exists c > 0, L, and n0 such that every n-vertex graph G = (V , E) that is η-far from being induced F-free, there exists a ℓ-vertex graph F ∈ F with ℓ ≤ L such that ★{F ≤ G} ≥ cnℓ . Remarks Short version: η-far from ind. F-free ⇒ “many” copies of a small F ∈ F Alon, Fischer, Krivelevich, and M. Szegedy ’00 for single graph F alternative proof by Lov´ asz and B. Szegedy ’04 R¨

• dl and S. obtained hypergraph generalization

proofs use iterated versions of the regularity lemma ❋♦r❜✐♥❞

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Generalized removal lemma

Theorem (Alon & Shapira ’05) For every η > 0 and every family of graphs F there exists c > 0, L, and n0 such that every n-vertex graph G = (V , E) that is η-far from being induced F-free, there exists a ℓ-vertex graph F ∈ F with ℓ ≤ L such that ★{F ≤ G} ≥ cnℓ . Remarks Short version: η-far from ind. F-free ⇒ “many” copies of a small F ∈ F Alon, Fischer, Krivelevich, and M. Szegedy ’00 for single graph F alternative proof by Lov´ asz and B. Szegedy ’04 R¨

• dl and S. obtained hypergraph generalization

proofs use iterated versions of the regularity lemma ❋♦r❜✐♥❞

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Generalized removal lemma

Theorem (Alon & Shapira ’05) For every η > 0 and every family of graphs F there exists c > 0, L, and n0 such that every n-vertex graph G = (V , E) that is η-far from being induced F-free, there exists a ℓ-vertex graph F ∈ F with ℓ ≤ L such that ★{F ≤ G} ≥ cnℓ . Remarks Short version: η-far from ind. F-free ⇒ “many” copies of a small F ∈ F Alon, Fischer, Krivelevich, and M. Szegedy ’00 for single graph F alternative proof by Lov´ asz and B. Szegedy ’04 R¨

• dl and S. obtained hypergraph generalization

proofs use iterated versions of the regularity lemma ⇒ decidable, hereditary hypergraph properties P = ❋♦r❜✐♥❞(F) are testable

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: ❋♦r❜✐♥❞ ★

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 ❋♦r❜✐♥❞ ★

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, ❋♦r❜✐♥❞ ★

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 ❋♦r❜✐♥❞ ★

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. ❋♦r❜✐♥❞ ★

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} ≥ cnvF for some F ∈ F with vF ≤ 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} ≥ cnvF for some F ∈ F with vF ≤ L ⇒ “easily” detectable by random sampling; q ∼ f (L/c)

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 (Gℓ)ℓ∈N with nℓ = |V (Gℓ)| → ∞ such that for every ℓ ∈ N ❋♦r❜✐♥❞ ★ ❋♦r❜✐♥❞

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 (Gℓ)ℓ∈N with nℓ = |V (Gℓ)| → ∞ such that for every ℓ ∈ N (i) Gℓ is η-far from ❋♦r❜✐♥❞(F) ★ ❋♦r❜✐♥❞

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 (Gℓ)ℓ∈N with nℓ = |V (Gℓ)| → ∞ such that for every ℓ ∈ N (i) Gℓ is η-far from ❋♦r❜✐♥❞(F) and (ii) every F ∈ F with vF ≤ ℓ ★ ❋♦r❜✐♥❞

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 (Gℓ)ℓ∈N with nℓ = |V (Gℓ)| → ∞ such that for every ℓ ∈ N (i) Gℓ is η-far from ❋♦r❜✐♥❞(F) and (ii) every F ∈ F with vF ≤ ℓ ★{F ≤ Gℓ} ≤ 1 ℓ nvF . ❋♦r❜✐♥❞

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 (Gℓ)ℓ∈N with nℓ = |V (Gℓ)| → ∞ such that for every ℓ ∈ N (i) Gℓ is η-far from ❋♦r❜✐♥❞(F) and (ii) every F ∈ F with vF ≤ ℓ ★{F ≤ Gℓ} ≤ 1 ℓ nvF . Idea (Gℓ)ℓ∈N contains convergent subsequence (G ′

ℓ)ℓ∈N with limit R

❋♦r❜✐♥❞

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 (Gℓ)ℓ∈N with nℓ = |V (Gℓ)| → ∞ such that for every ℓ ∈ N (i) Gℓ is η-far from ❋♦r❜✐♥❞(F) and (ii) every F ∈ F with vF ≤ ℓ ★{F ≤ Gℓ} ≤ 1 ℓ nvF . Idea (Gℓ)ℓ∈N contains convergent subsequence (G ′

ℓ)ℓ∈N with limit R

R must be ind. F-free, since t(F, Gℓ) → 0 for all F ∈ F ❋♦r❜✐♥❞

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 (Gℓ)ℓ∈N with nℓ = |V (Gℓ)| → ∞ such that for every ℓ ∈ N (i) Gℓ is η-far from ❋♦r❜✐♥❞(F) and (ii) every F ∈ F with vF ≤ ℓ ★{F ≤ Gℓ} ≤ 1 ℓ nvF . Idea (Gℓ)ℓ∈N contains convergent subsequence (G ′

ℓ)ℓ∈N with limit R

R must be ind. F-free, since t(F, Gℓ) → 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 (Gℓ)ℓ∈N with nℓ = |V (Gℓ)| → ∞ such that for every ℓ ∈ N (i) Gℓ is η-far from ❋♦r❜✐♥❞(F) and (ii) every F ∈ F with vF ≤ ℓ ★{F ≤ Gℓ} ≤ 1 ℓ nvF . Idea (Gℓ)ℓ∈N contains convergent subsequence (G ′

ℓ)ℓ∈N with limit R

R must be ind. F-free, since t(F, Gℓ) → 0 for all F ∈ F ⇒ “R has P = ❋♦r❜✐♥❞(F)” But all Gℓ 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 (Gℓ)ℓ∈N with nℓ = |V (Gℓ)| → ∞ such that for every ℓ ∈ N (i) Gℓ is η-far from ❋♦r❜✐♥❞(F) and (ii) every F ∈ F with vF ≤ ℓ ★{F ≤ Gℓ} ≤ 1 ℓ nvF . Idea (Gℓ)ℓ∈N contains convergent subsequence (G ′

ℓ)ℓ∈N with limit R

R must be ind. F-free, since t(F, Gℓ) → 0 for all F ∈ F ⇒ “R has P = ❋♦r❜✐♥❞(F)” But all Gℓ 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 (Gℓ)ℓ∈N there exist an infinite subsequence (G ′

ℓ)ℓ∈N, a sequence of symmetric, measurable functions

(Rℓ)ℓ∈N with Rℓ: [0, 1]2 → [0, 1] 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) Testing, Hypergraph Regularity November 2008

Limits of graph sequences

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

ℓ)ℓ∈N, a sequence of symmetric, measurable functions

(Rℓ)ℓ∈N with Rℓ: [0, 1]2 → [0, 1] 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 ✮ limℓ→∞ d(G ′

ℓ, Rℓ) = 0, and

✭iii ✮ limℓ→∞ d(Rℓ, R) = 0, where for two functions A, B : [0, 1]2 → [0, 1] we set d(A, B) sup

U⊆[0,1]

• U×U

|A(x, y) − B(x, y)|❞x❞y .

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Limits of graph sequences

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

ℓ)ℓ∈N, a sequence of symmetric, measurable functions

(Rℓ)ℓ∈N with Rℓ: [0, 1]2 → [0, 1] 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 ✮ limℓ→∞ d(G ′

ℓ, Rℓ) = 0, and

✭iii ✮ limℓ→∞ d(Rℓ, R) = 0, where for two functions A, B : [0, 1]2 → [0, 1] we set d(A, B) sup

U⊆[0,1]

• U×U

|A(x, y) − B(x, y)|❞x❞y .

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Sketch

Proof. let n be sufficiently large and split [0,1] into n intervals Z1, . . . , Zn 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 Z1, . . . , Zn of length 1/n pick zi ∈ Zi uniformly at random and define Rz and Hz

n

Rz/Hz

n(x, y) =

     1 if (x, y) ∈ Zi × Zj and R(zi, zj) = 1 , if (x, y) ∈ Zi × Zj and R(zi, zj) = 0 , R(x, y)/G ′

n(x, y)

• therwise .

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Sketch

Proof. let n be sufficiently large and split [0,1] into n intervals Z1, . . . , Zn of length 1/n pick zi ∈ Zi uniformly at random and define Rz and Hz

n

Rz/Hz

n(x, y) =

     1 if (x, y) ∈ Zi × Zj and R(zi, zj) = 1 , if (x, y) ∈ Zi × Zj and R(zi, zj) = 0 , R(x, y)/G ′

n(x, y)

• therwise .

Fact 1 With prob. 1 we have t(F, Hz

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 Z1, . . . , Zn of length 1/n pick zi ∈ Zi uniformly at random and define Rz and Hz

n

Rz/Hz

n(x, y) =

     1 if (x, y) ∈ Zi × Zj and R(zi, zj) = 1 , if (x, y) ∈ Zi × Zj and R(zi, zj) = 0 , R(x, y)/G ′

n(x, y)

• therwise .

Fact 1 With prob. 1 we have t(F, Hz

n) = 0 for all F ∈ F

⇒ Hz

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 Z1, . . . , Zn of length 1/n pick zi ∈ Zi uniformly at random and define Rz and Hz

n

Rz/Hz

n(x, y) =

     1 if (x, y) ∈ Zi × Zj and R(zi, zj) = 1 , if (x, y) ∈ Zi × Zj and R(zi, zj) = 0 , R(x, y)/G ′

n(x, y)

• therwise .

Fact 1 With prob. 1 we have t(F, Hz

n) = 0 for all F ∈ F

⇒ Hz

n ∈ P

⋆ Fact 2 W.h.p. d(Rz, 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 Z1, . . . , Zn of length 1/n pick zi ∈ Zi uniformly at random and define Rz and Hz

n

Rz/Hz

n(x, y) =

     1 if (x, y) ∈ Zi × Zj and R(zi, zj) = 1 , if (x, y) ∈ Zi × Zj and R(zi, zj) = 0 , R(x, y)/G ′

n(x, y)

• therwise .

Fact 1 With prob. 1 we have t(F, Hz

n) = 0 for all F ∈ F

⇒ Hz

n ∈ P

⋆ Fact 2 W.h.p. d(Rz, R) = o(1). ⇒ d(Hz

n, Rz) Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Proof — Sketch

Proof. let n be sufficiently large and split [0,1] into n intervals Z1, . . . , Zn of length 1/n pick zi ∈ Zi uniformly at random and define Rz and Hz

n

Rz/Hz

n(x, y) =

     1 if (x, y) ∈ Zi × Zj and R(zi, zj) = 1 , if (x, y) ∈ Zi × Zj and R(zi, zj) = 0 , R(x, y)/G ′

n(x, y)

• therwise .

Fact 1 With prob. 1 we have t(F, Hz

n) = 0 for all F ∈ F

⇒ Hz

n ∈ P

⋆ Fact 2 W.h.p. d(Rz, R) = o(1). ⇒ d(Hz

n, Rz) ≤ 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 Z1, . . . , Zn of length 1/n pick zi ∈ Zi uniformly at random and define Rz and Hz

n

Rz/Hz

n(x, y) =

     1 if (x, y) ∈ Zi × Zj and R(zi, zj) = 1 , if (x, y) ∈ Zi × Zj and R(zi, zj) = 0 , R(x, y)/G ′

n(x, y)

• therwise .

Fact 1 With prob. 1 we have t(F, Hz

n) = 0 for all F ∈ F

⇒ Hz

n ∈ P

⋆ Fact 2 W.h.p. d(Rz, R) = o(1). ⇒ d(Hz

n, Rz) ≤ d(G ′ n, R) ≤ d(G ′ n, Rn) + d(Rn, 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 Z1, . . . , Zn of length 1/n pick zi ∈ Zi uniformly at random and define Rz and Hz

n

Rz/Hz

n(x, y) =

     1 if (x, y) ∈ Zi × Zj and R(zi, zj) = 1 , if (x, y) ∈ Zi × Zj and R(zi, zj) = 0 , R(x, y)/G ′

n(x, y)

• therwise .

Fact 1 With prob. 1 we have t(F, Hz

n) = 0 for all F ∈ F

⇒ Hz

n ∈ P

⋆ Fact 2 W.h.p. d(Rz, R) = o(1). ⇒ d(Hz

n, Rz) ≤ d(G ′ n, R) ≤ d(G ′ n, Rn) + d(Rn, 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 Z1, . . . , Zn of length 1/n pick zi ∈ Zi uniformly at random and define Rz and Hz

n

Rz/Hz

n(x, y) =

     1 if (x, y) ∈ Zi × Zj and R(zi, zj) = 1 , if (x, y) ∈ Zi × Zj and R(zi, zj) = 0 , R(x, y)/G ′

n(x, y)

• therwise .

Fact 1 With prob. 1 we have t(F, Hz

n) = 0 for all F ∈ F

⇒ Hz

n ∈ P

⋆ Fact 2 W.h.p. d(Rz, R) = o(1). ⇒ d(Hz

n, Rz) ≤ d(G ′ n, R) ≤ d(G ′ n, Rn) + d(Rn, R) = o(1)

⇒ d(Hz

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 Z1, . . . , Zn of length 1/n pick zi ∈ Zi uniformly at random and define Rz and Hz

n

Rz/Hz

n(x, y) =

     1 if (x, y) ∈ Zi × Zj and R(zi, zj) = 1 , if (x, y) ∈ Zi × Zj and R(zi, zj) = 0 , R(x, y)/G ′

n(x, y)

• therwise .

Fact 1 With prob. 1 we have t(F, Hz

n) = 0 for all F ∈ F

⇒ Hz

n ∈ P

⋆ Fact 2 W.h.p. d(Rz, R) = o(1). ⇒ d(Hz

n, Rz) ≤ d(G ′ n, R) ≤ d(G ′ n, Rn) + d(Rn, R) = o(1)

⇒ d(Hz

n, G ′ n) ≤ d(Hz, Rz) + d(Rz, R) + d(R, 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 Z1, . . . , Zn of length 1/n pick zi ∈ Zi uniformly at random and define Rz and Hz

n

Rz/Hz

n(x, y) =

     1 if (x, y) ∈ Zi × Zj and R(zi, zj) = 1 , if (x, y) ∈ Zi × Zj and R(zi, zj) = 0 , R(x, y)/G ′

n(x, y)

• therwise .

Fact 1 With prob. 1 we have t(F, Hz

n) = 0 for all F ∈ F

⇒ Hz

n ∈ P

⋆ Fact 2 W.h.p. d(Rz, R) = o(1). ⇒ d(Hz

n, Rz) ≤ d(G ′ n, R) ≤ d(G ′ n, Rn) + d(Rn, R) = o(1)

⇒ d(Hz

n, G ′ n) ≤ d(Hz, Rz) + d(Rz, R) + d(R, 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 Z1, . . . , Zn of length 1/n pick zi ∈ Zi uniformly at random and define Rz and Hz

n

Rz/Hz

n(x, y) =

     1 if (x, y) ∈ Zi × Zj and R(zi, zj) = 1 , if (x, y) ∈ Zi × Zj and R(zi, zj) = 0 , R(x, y)/G ′

n(x, y)

• therwise .

Fact 1 With prob. 1 we have t(F, Hz

n) = 0 for all F ∈ F

⇒ Hz

n ∈ P

⋆ Fact 2 W.h.p. d(Rz, R) = o(1). ⇒ d(Hz

n, Rz) ≤ d(G ′ n, R) ≤ d(G ′ n, Rn) + d(Rn, R) = o(1)

⇒ d(Hz

n, G ′ n) ≤ d(Hz, Rz) + d(Rz, R) + d(R, G ′ n) = o(1)

⇒ Hz 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 |e(U) − d |U|

2

• | = o(n2) for all U ⊆ V .

★

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 |e(U) − d |U|

2

• | = o(n2) for all U ⊆ V .

Definition (dev) A graph G = (V , E) of density d satisfies dev if

• u0,u1∈Vn
• v0,v1∈Vn
• i∈{0,1}
• j∈{0,1}

g(ui, vj) = o(n4) , 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 |e(U) − d |U|

2

• | = o(n2) for all U ⊆ V .

Definition (dev) A graph G = (V , E) of density d satisfies dev if

• u0,u1∈Vn
• v0,v1∈Vn
• i∈{0,1}
• j∈{0,1}

g(ui, vj) = o(n4) , 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 ★{C4 ⊆ G} ≤ d4n4 + o(n4) .

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

• f 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

• f 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 |e(U) − d |U|

3

• | = o(n3) for all U ⊆ V .

★ ★

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 |e(U) − d |U|

3

• | = o(n3) for all U ⊆ V .

Definition (oct) A 3-uniform hypergraph H = (V , E) of density d satisfies oct if ★{K (3)

2,2,2 ⊆ H} ≤ d8n6 + o(n6) .

★

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 |e(U) − d |U|

3

• | = o(n3) for all U ⊆ V .

Definition (oct) A 3-uniform hypergraph H = (V , E) of density d satisfies oct if ★{K (3)

2,2,2 ⊆ H} ≤ d8n6 + o(n6) .

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 |e(U) − d |U|

3

• | = o(n3) for all U ⊆ V .

Definition (oct) A 3-uniform hypergraph H = (V , E) of density d satisfies oct if ★{K (3)

2,2,2 ⊆ H} ≤ d8n6 + o(n6) .

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 |e(U) − d |U|

3

• | = o(n3) for all U ⊆ V .

Definition (oct) A 3-uniform hypergraph H = (V , E) of density d satisfies oct if ★{K (3)

2,2,2 ⊆ H} ≤ d8n6 + o(n6) .

Fact There exists a 3-uniform hypergraph H = (V , E) of density 1/8 ± o(1) such that H satisfies weak-disc, but ★{K (3)

2,2,2 ⊆ H} ≥ (1 − o(1))( 1 2)12n6 = (1 − o(1))( 1 8)4n6 > (1 − o(1))( 1 8)8n6.

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

• |E ∩ K3(G)| − d|K3(G)|
• = o(n3) for all graphs G with vertex set V .

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

• |E ∩ K3(G)| − d|K3(G)|
• = o(n3) for all graphs G with vertex set V .

Definition (dev) A 3-uniform hypergraph H = (V , E) of density d satisfies dev if

• u0,u1
• v0,v1
• w0,w1
• i,j,k∈{0,1}

h(ui, vj, wk) = o(n6) , 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

• |E ∩ K3(G)| − d|K3(G)|
• = o(n3) for all graphs G with vertex set V .

Definition (dev) A 3-uniform hypergraph H = (V , E) of density d satisfies dev if

• u0,u1
• v0,v1
• w0,w1
• i,j,k∈{0,1}

h(ui, vj, wk) = o(n6) , 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

• |E ∩ K3(G)| − d|K3(G)|
• = o(n3) for all graphs G with vertex set V .

Definition (dev) A 3-uniform hypergraph H = (V , E) of density d satisfies dev if

• u0,u1
• v0,v1
• w0,w1
• i,j,k∈{0,1}

h(ui, vj, wk) = o(n6) , 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 , EG) of density d2 satisfying disc / dev / oct ★

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Third attempt

Set-Up a graph G = (V , EG) of density d2 satisfying disc / dev / oct a 3-uniform hypergraph H = (V , EH)⊆K3(G) with relative density d3 ★

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Third attempt

Set-Up a graph G = (V , EG) of density d2 satisfying disc / dev / oct a 3-uniform hypergraph H = (V , EH)⊆K3(G) with relative density d3 Definition (disc) (G, H) satisfies disc if

• |E ∩ K3(G ′)| − d3|K3(G ′)|
• = o(n3) for all subgraphs G ′ ⊆ G .

★

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Third attempt

Set-Up a graph G = (V , EG) of density d2 satisfying disc / dev / oct a 3-uniform hypergraph H = (V , EH)⊆K3(G) with relative density d3 Definition (disc) (G, H) satisfies disc if

• |E ∩ K3(G ′)| − d3|K3(G ′)|
• = o(n3) for all subgraphs G ′ ⊆ G .

Definition (oct) (G, H) satisfies oct if ★{K (3)

2,2,2 ⊆ H} ≤ d8 3d12 2 n6 + o(n6) .

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Third attempt (cont’d)

Set-Up a graph G = (V , EG) of density d2 satisfying disc / dev / oct a 3-uniform hypergraph H = (V , EH)⊆K3(G) with relative density d3 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 , EG) of density d2 satisfying disc / dev / oct a 3-uniform hypergraph H = (V , EH)⊆K3(G) with relative density d3 Definition (dev) (G, H) satisfies dev if

• u0,u1
• v0,v1
• w0,w1
• i,j,k∈{0,1}

h(ui, vj, wk) = o(n6) , where h(u, v, w) =        1 − d if {u, v, w} ∈ EH, −d if {u, v, w} ∈ K3(G) \ EH,

• therwise.

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Third attempt (cont’d)

Set-Up a graph G = (V , EG) of density d2 satisfying disc / dev / oct a 3-uniform hypergraph H = (V , EH)⊆K3(G) with relative density d3 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 , EG) of density d2 satisfying disc / dev / oct a 3-uniform hypergraph H = (V , EH)⊆K3(G) with relative density d3 Remarks disc, dev, and oct are equivalent (similar to Chung & Graham 1990) but the error-terms are only “useful” if ε ≪ d2, i.e.,

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Third attempt (cont’d)

Set-Up a graph G = (V , EG) of density d2 satisfying disc / dev / oct a 3-uniform hypergraph H = (V , EH)⊆K3(G) with relative density d3 Remarks disc, dev, and oct are equivalent (similar to Chung & Graham 1990) but the error-terms are only “useful” if ε ≪ d2, i.e.,

disc : o(n3) ≪ d3d3

2n3

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Third attempt (cont’d)

Set-Up a graph G = (V , EG) of density d2 satisfying disc / dev / oct a 3-uniform hypergraph H = (V , EH)⊆K3(G) with relative density d3 Remarks disc, dev, and oct are equivalent (similar to Chung & Graham 1990) but the error-terms are only “useful” if ε ≪ d2, i.e.,

disc : o(n3) ≪ d3d3

2n3

• ct : o(n6) ≪ d8

3d12 2 n6

dev : o(n6) ≪ d8

3d12 2 n6

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Third attempt (cont’d)

Set-Up a graph G = (V , EG) of density d2 satisfying disc / dev / oct a 3-uniform hypergraph H = (V , EH)⊆K3(G) with relative density d3 Remarks disc, dev, and oct are equivalent (similar to Chung & Graham 1990) but the error-terms are only “useful” if ε ≪ d2, i.e.,

disc : o(n3) ≪ d3d3

2n3

• ct : o(n6) ≪ d8

3d12 2 n6

dev : o(n6) ≪ d8

3d12 2 n6

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., d2) 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 , EG) of density d2 satisfying disc(ε2)/ dev(ε2)/ oct(ε2) a 3-uniform hypergraph H = (V , EH)⊆K3(G) with relative density d3 ★

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Final attempt

Set-Up a graph G = (V , EG) of density d2 satisfying disc(ε2)/ dev(ε2)/ oct(ε2) a 3-uniform hypergraph H = (V , EH)⊆K3(G) with relative density d3 d3 ≫ ε3 ≫ d2 ≫ ε2 ★

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Final attempt

Set-Up a graph G = (V , EG) of density d2 satisfying disc(ε2)/ dev(ε2)/ oct(ε2) a 3-uniform hypergraph H = (V , EH)⊆K3(G) with relative density d3 d3 ≫ ε3 ≫ d2 ≫ ε2 Definition (disc) (G, H) satisfies disc(ε3) if

• |E ∩ K3(G ′)| − d3|K3(G ′)|
• ≤ ε3d3

2n3 for all subgraphs G ′ ⊆ G .

★

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Final attempt

Set-Up a graph G = (V , EG) of density d2 satisfying disc(ε2)/ dev(ε2)/ oct(ε2) a 3-uniform hypergraph H = (V , EH)⊆K3(G) with relative density d3 d3 ≫ ε3 ≫ d2 ≫ ε2 Definition (disc) (G, H) satisfies disc(ε3) if

• |E ∩ K3(G ′)| − d3|K3(G ′)|
• ≤ ε3d3

2n3 for all subgraphs G ′ ⊆ G .

Definition (oct) (G, H) satisfies oct(ε3) if ★{K (3)

2,2,2 ⊆ H} ≤ d8 3d12 2 n6 + ε3d12 2 n6 .

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Final attempt (cont’d)

Set-Up a graph G = (V , EG) of density d2 satisfying disc(ε2)/ dev(ε2)/ oct(ε2) a 3-uniform hypergraph H = (V , EH)⊆K3(G) with relative density d3 d3 ≫ ε3 ≫ d2 ≫ ε2 Definition (dev) (G, H) satisfies dev(ε3) if

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

Quasi-random hypergraphs: Final attempt (cont’d)

Set-Up a graph G = (V , EG) of density d2 satisfying disc(ε2)/ dev(ε2)/ oct(ε2) a 3-uniform hypergraph H = (V , EH)⊆K3(G) with relative density d3 d3 ≫ ε3 ≫ d2 ≫ ε2 Definition (dev) (G, H) satisfies dev(ε3) if

• u0,u1
• v0,v1
• w0,w1
• i,j,k∈{0,1}

h(ui, vj, wk) ≤ ε3d12

2 n6 ,

where h(u, v, w) =        1 − d if {u, v, w} ∈ EH, −d if {u, v, w} ∈ K3(G) \ EH,

• therwise.

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¨

• dl (disc), Gowers (dev), and

Haxel–Nagle–R¨

• dl (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¨

• dl (disc), Gowers (dev), and

Haxel–Nagle–R¨

• dl (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¨

• dl (disc), Gowers (dev), and

Haxel–Nagle–R¨

• dl (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¨

• dl, 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¨

• dl, S.)

disc, dev, and oct are equivalent for 3-uniform hypergraphs. I.e., For all d3, ε3 there exists δ3 such that for all d2, ε2 there exists δ2 and n0 such that

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

New Result

Theorem (Nagle, Poerschke, R¨

• dl, S.)

disc, dev, and oct are equivalent for 3-uniform hypergraphs. I.e., For all d3, ε3 there exists δ3 such that for all d2, ε2 there exists δ2 and n0 such that if (H, G) with densities d3 and d2 satisfies disc(δ2, δ3),

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

New Result

Theorem (Nagle, Poerschke, R¨

• dl, S.)

disc, dev, and oct are equivalent for 3-uniform hypergraphs. I.e., For all d3, ε3 there exists δ3 such that for all d2, ε2 there exists δ2 and n0 such that if (H, G) with densities d3 and d2 satisfies disc(δ2, δ3), then (H, G) must satisfy dev(ε2, ε3); . . .

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

New Result

Theorem (Nagle, Poerschke, R¨

• dl, S.)

disc, dev, and oct are equivalent for 3-uniform hypergraphs. I.e., For all d3, ε3 there exists δ3 such that for all d2, ε2 there exists δ2 and n0 such that if (H, G) with densities d3 and d2 satisfies disc(δ2, δ3), then (H, G) must satisfy dev(ε2, ε3); . . . Remarks main part of the proof: disc ⇒ dev / oct

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008

New Result

Theorem (Nagle, Poerschke, R¨

• dl, S.)

disc, dev, and oct are equivalent for 3-uniform hypergraphs. I.e., For all d3, ε3 there exists δ3 such that for all d2, ε2 there exists δ2 and n0 such that if (H, G) with densities d3 and d2 satisfies disc(δ2, δ3), then (H, G) must satisfy dev(ε2, ε3); . . . 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¨

• dl, S.)

disc, dev, and oct are equivalent for 3-uniform hypergraphs. I.e., For all d3, ε3 there exists δ3 such that for all d2, ε2 there exists δ2 and n0 such that if (H, G) with densities d3 and d2 satisfies disc(δ2, δ3), then (H, G) must satisfy dev(ε2, ε3); . . . Remarks main part of the proof: disc ⇒ dev / oct proof is based on two applications of the hypergraph regularity lemma

• ther 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¨

• dl is not equivalent to

disc / dev / 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 stronger version of disc due to Frankl–R¨

• dl is not equivalent to

disc / dev / oct Algorithmic regularity lemma for all three versions of regularity

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¨

• dl is not equivalent to

disc / dev / oct Algorithmic regularity lemma for all three versions of regularity Open Problems generalizations for arbitrary k

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¨

• dl is not equivalent to

disc / dev / oct Algorithmic regularity lemma for all three versions of regularity Open Problems generalizations for arbitrary k (work in progress)

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¨

• dl is not equivalent to

disc / dev / oct Algorithmic regularity lemma for all three versions of regularity Open Problems generalizations for arbitrary k (work in progress) Is there a polynomial dependence possible for disc(δ2, δ3) ⇒ dev(ε2, ε3)?

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¨

• dl is not equivalent to

disc / dev / oct Algorithmic regularity lemma for all three versions of regularity Open Problems generalizations for arbitrary k (work in progress) Is there a polynomial dependence possible for disc(δ2, δ3) ⇒ dev(ε2, ε3)? I.e. does δ3 = εC

3 for some C suffice?

Mathias Schacht (HU-Berlin) Testing, Hypergraph Regularity November 2008