On the Regularity Method for Hypergraphs Mathias Schacht October - - PowerPoint PPT Presentation

On the Regularity Method for Hypergraphs

Mathias Schacht October 2004

Regularity Method for Hypergraphs

1

Outline

1 Density Theorems Szemerédi’s Density Theorem Density Theorems of Furstenberg and Katznelson

Regularity Method for Hypergraphs

1

Outline

1 Density Theorems Szemerédi’s Density Theorem Density Theorems of Furstenberg and Katznelson 2 An Extremal Hypergraph problem Connection to the Density Theorems Szemeredi’s Regularity Lemma for Graphs Solution of the Extremal Problem for Graphs

Regularity Method for Hypergraphs

1

Outline

1 Density Theorems Szemerédi’s Density Theorem Density Theorems of Furstenberg and Katznelson 2 An Extremal Hypergraph problem Connection to the Density Theorems Szemeredi’s Regularity Lemma for Graphs Solution of the Extremal Problem for Graphs 3 Regularity Method for Hypergraphs History Regularity Lemma Counting Lemma

Regularity Method for Hypergraphs Density Theorems

2

Arithmetic Progressions

Theorem (van der Waerden 1927). For all positive integers k and s there exist an n0 such that every s-colouring of [n] = {1, . . . , n} (n ≥ n0) contains a monochromatic AP(k), i.e., a monochromatic arithmetic pro- gression of length k.

Regularity Method for Hypergraphs Density Theorems

2

Arithmetic Progressions

Theorem (van der Waerden 1927). For all positive integers k and s there exist an n0 such that every s-colouring of [n] = {1, . . . , n} (n ≥ n0) contains a monochromatic AP(k), i.e., a monochromatic arithmetic pro- gression of length k. Question (Erd˝

• s & Turán 1936). Let rk(n) be the maximal size of an

AP(k)-free subset Z of [n]. Is rk(n) = o(n)?

Regularity Method for Hypergraphs Density Theorems

2

Arithmetic Progressions

Theorem (van der Waerden 1927). For all positive integers k and s there exist an n0 such that every s-colouring of [n] = {1, . . . , n} (n ≥ n0) contains a monochromatic AP(k), i.e., a monochromatic arithmetic pro- gression of length k. Question (Erd˝

• s & Turán 1936). Let rk(n) be the maximal size of an

AP(k)-free subset Z of [n]. Is rk(n) = o(n)? Theorem (Roth 1954). r3(n) = o(n) Theorem (Szemerédi 1969). r4(n) = o(n)

Regularity Method for Hypergraphs Density Theorems

2

Arithmetic Progressions

Theorem (van der Waerden 1927). For all positive integers k and s there exist an n0 such that every s-colouring of [n] = {1, . . . , n} (n ≥ n0) contains a monochromatic AP(k), i.e., a monochromatic arithmetic pro- gression of length k. Question (Erd˝

• s & Turán 1936). Let rk(n) be the maximal size of an

AP(k)-free subset Z of [n]. Is rk(n) = o(n)? Theorem (Roth 1954). r3(n) = o(n) Theorem (Szemerédi 1969). r4(n) = o(n) Theorem (Szemerédi 1975). For every positive integer k rk(n) = o(n) .

Regularity Method for Hypergraphs Density Theorems

2

Arithmetic Progressions

Theorem (van der Waerden 1927). For all positive integers k and s there exist an n0 such that every s-colouring of [n] = {1, . . . , n} (n ≥ n0) contains a monochromatic AP(k), i.e., a monochromatic arithmetic pro- gression of length k. Question (Erd˝

• s & Turán 1936). Let rk(n) be the maximal size of an

AP(k)-free subset Z of [n]. Is rk(n) = o(n)? Theorem (Roth 1954). r3(n) = o(n) Theorem (Szemerédi 1969). r4(n) = o(n) Theorem (Szemerédi 1975). For every positive integer k rk(n) = o(n) . Alternative proofs: Furstenberg (1977)

Regularity Method for Hypergraphs Density Theorems

2

Arithmetic Progressions

Theorem (van der Waerden 1927). For all positive integers k and s there exist an n0 such that every s-colouring of [n] = {1, . . . , n} (n ≥ n0) contains a monochromatic AP(k), i.e., a monochromatic arithmetic pro- gression of length k. Question (Erd˝

• s & Turán 1936). Let rk(n) be the maximal size of an

AP(k)-free subset Z of [n]. Is rk(n) = o(n)? Theorem (Roth 1954). r3(n) = o(n) Theorem (Szemerédi 1969). r4(n) = o(n) Theorem (Szemerédi 1975). For every positive integer k rk(n) = o(n) . Alternative proofs: Furstenberg (1977), Gowers (2001)

Regularity Method for Hypergraphs Density Theorems

2

Arithmetic Progressions

Theorem (van der Waerden 1927). For all positive integers k and s there exist an n0 such that every s-colouring of [n] = {1, . . . , n} (n ≥ n0) contains a monochromatic AP(k), i.e., a monochromatic arithmetic pro- gression of length k. Question (Erd˝

• s & Turán 1936). Let rk(n) be the maximal size of an

AP(k)-free subset Z of [n]. Is rk(n) = o(n)? Theorem (Roth 1954). r3(n) = o(n) Theorem (Szemerédi 1969). r4(n) = o(n) Theorem (Szemerédi 1975). For every positive integer k rk(n) = o(n) . Alternative proofs: Furstenberg (1977), Gowers (2001), Tao (2004++)

Regularity Method for Hypergraphs Density Theorems

3

Multidimensional versions of Szemerédi’s Theorem

Question (Erd˝

• s & Graham 1970). Given a finite configuration C in Zd.

Let rC(n) be the maximal size of a subset Z of [n]d not containing a ho- mothetic copy of C.

Regularity Method for Hypergraphs Density Theorems

3

Multidimensional versions of Szemerédi’s Theorem

Question (Erd˝

• s & Graham 1970). Given a finite configuration C in Zd.

Let rC(n) be the maximal size of a subset Z of [n]d not containing a ho- mothetic copy of C. Is rC(n) = o(n2) if C = {(0, 0), (1, 0), (0, 1), (1, 1)} is a square?

Regularity Method for Hypergraphs Density Theorems

3

Multidimensional versions of Szemerédi’s Theorem

Question (Erd˝

• s & Graham 1970). Given a finite configuration C in Zd.

Let rC(n) be the maximal size of a subset Z of [n]d not containing a ho- mothetic copy of C. Is rC(n) = o(n2) if C = {(0, 0), (1, 0), (0, 1), (1, 1)} is a square? Theorem (Ajtai & Szemerédi 1974). If C = {(0, 0), (1, 0), (0, 1)} is an isosceles right triangle, then rC(n) = o(n2).

Regularity Method for Hypergraphs Density Theorems

3

Multidimensional versions of Szemerédi’s Theorem

Question (Erd˝

• s & Graham 1970). Given a finite configuration C in Zd.

Let rC(n) be the maximal size of a subset Z of [n]d not containing a ho- mothetic copy of C. Is rC(n) = o(n2) if C = {(0, 0), (1, 0), (0, 1), (1, 1)} is a square? Theorem (Ajtai & Szemerédi 1974). If C = {(0, 0), (1, 0), (0, 1)} is an isosceles right triangle, then rC(n) = o(n2). Theorem (Furstenberg & Katznelson 1978). For every finite configura- tion C in Zd rC(n) = o(nd) .

Regularity Method for Hypergraphs Density Theorems

4

Other Density Theorems

Theorem (Furstenberg & Katznelson 1985). If Z ⊂ Fn

q does not contain

an affine subspace of dimension d, then |Z| = o(qn) .

Regularity Method for Hypergraphs Density Theorems

4

Other Density Theorems

Theorem (Furstenberg & Katznelson 1985). If Z ⊂ Fn

q does not contain

an affine subspace of dimension d, then |Z| = o(qn) . Theorem (Furstenberg & Katznelson 1985). Let G be a finite abelian

• group. If Z ⊂ Gn does not contain a coset of a subgroup of Gn isomorphic

to G, then |Z| = o(|G|n) .

Regularity Method for Hypergraphs Density Theorems

4

Other Density Theorems

Theorem (Furstenberg & Katznelson 1985). If Z ⊂ Fn

q does not contain

an affine subspace of dimension d, then |Z| = o(qn) . Theorem (Furstenberg & Katznelson 1985). Let G be a finite abelian

• group. If Z ⊂ Gn does not contain a coset of a subgroup of Gn isomorphic

to G, then |Z| = o(|G|n) . “Theme of this talk”: New combinatorial proofs of the Density Theorems mentioned above.

Regularity Method for Hypergraphs Density Theorems

4

Other Density Theorems

Theorem (Furstenberg & Katznelson 1985). If Z ⊂ Fn

q does not contain

an affine subspace of dimension d, then |Z| = o(qn) . Theorem (Furstenberg & Katznelson 1985). Let G be a finite abelian

• group. If Z ⊂ Gn does not contain a coset of a subgroup of Gn isomorphic

to G, then |Z| = o(|G|n) . “Theme of this talk”: New combinatorial proofs of the Density Theorems mentioned above. Remark.

• density version of the Hales–Jewett Theorem

− → Furstenberg & Katznelson 1991

Regularity Method for Hypergraphs Density Theorems

4

Other Density Theorems

Theorem (Furstenberg & Katznelson 1985). If Z ⊂ Fn

q does not contain

an affine subspace of dimension d, then |Z| = o(qn) . Theorem (Furstenberg & Katznelson 1985). Let G be a finite abelian

• group. If Z ⊂ Gn does not contain a coset of a subgroup of Gn isomorphic

to G, then |Z| = o(|G|n) . “Theme of this talk”: New combinatorial proofs of the Density Theorems mentioned above. Remark.

• density version of the Hales–Jewett Theorem

− → Furstenberg & Katznelson 1991

• polynomial extensions

− → Bergelson & Leibman 1996, Bergelson & McCutcheon 2000

Regularity Method for Hypergraphs

5

Review/Outline

1 Density Theorems Szemerédi’s Density Theorem Density Theorems of Furstenberg and Katznelson

Regularity Method for Hypergraphs

5

Review/Outline

1 Density Theorems Szemerédi’s Density Theorem Density Theorems of Furstenberg and Katznelson 2 An Extremal Hypergraph problem Connection to the Density Theorems Szemeredi’s Regularity Lemma for Graphs Solution of the Extremal Problem for Graphs 3 Regularity Method for Hypergraphs History Regularity Lemma Counting Lemma

Regularity Method for Hypergraphs An Extremal Problem

6

An Extremal Hypergraph Problem

Theorem (Gowers 2003+, Nagle, Rödl, S. & Skokan 2003+). If H(k) is a k-uniform hypergraph on n vertices such that every edge of H(k) is contained in precisely one K(k)

k+1, then

Regularity Method for Hypergraphs An Extremal Problem

6

An Extremal Hypergraph Problem

Theorem (Gowers 2003+, Nagle, Rödl, S. & Skokan 2003+). If H(k) is a k-uniform hypergraph on n vertices such that every edge of H(k) is contained in precisely one K(k)

k+1, then

|H(k)| = o(nk) .

Regularity Method for Hypergraphs An Extremal Problem

6

An Extremal Hypergraph Problem

Theorem (Gowers 2003+, Nagle, Rödl, S. & Skokan 2003+). If H(k) is a k-uniform hypergraph on n vertices such that every edge of H(k) is contained in precisely one K(k)

k+1, then

|H(k)| = o(nk) . k = 2 − → Ruzsa & Szemerédi 1978

Regularity Method for Hypergraphs An Extremal Problem

6

An Extremal Hypergraph Problem

Theorem (Gowers 2003+, Nagle, Rödl, S. & Skokan 2003+). If H(k) is a k-uniform hypergraph on n vertices such that every edge of H(k) is contained in precisely one K(k)

k+1, then

|H(k)| = o(nk) . k = 2 − → Ruzsa & Szemerédi 1978 k = 3 − → Frankl & Rödl 2002

Regularity Method for Hypergraphs An Extremal Problem

6

An Extremal Hypergraph Problem

Theorem (Gowers 2003+, Nagle, Rödl, S. & Skokan 2003+). If H(k) is a k-uniform hypergraph on n vertices such that every edge of H(k) is contained in precisely one K(k)

k+1, then

|H(k)| = o(nk) . k = 2 − → Ruzsa & Szemerédi 1978 k = 3 − → Frankl & Rödl 2002 k = 4 − → Rödl & Skokan 2002+

Regularity Method for Hypergraphs An Extremal Problem

7

Solymosi: Ruzsa–Szemerédi ⇒ Ajtai–Szemerédi

• Let Z ⊆ [n]2 not containing a homothetic copy of an isosceles triangle

Regularity Method for Hypergraphs An Extremal Problem

7

Solymosi: Ruzsa–Szemerédi ⇒ Ajtai–Szemerédi

• Let Z ⊆ [n]2 not containing a homothetic copy of an isosceles triangle

Consider the tripartite graph G = (V, E) with V = Vhori ∪ Vvert ∪ Vdiag

Regularity Method for Hypergraphs An Extremal Problem

7

Solymosi: Ruzsa–Szemerédi ⇒ Ajtai–Szemerédi

• Let Z ⊆ [n]2 not containing a homothetic copy of an isosceles triangle

Consider the tripartite graph G = (V, E) with V = Vhori ∪ Vvert ∪ Vdiag where: Vhori = set of horizontal lines

• |V | = n

Regularity Method for Hypergraphs An Extremal Problem

7

Solymosi: Ruzsa–Szemerédi ⇒ Ajtai–Szemerédi

• Let Z ⊆ [n]2 not containing a homothetic copy of an isosceles triangle

Consider the tripartite graph G = (V, E) with V = Vhori ∪ Vvert ∪ Vdiag where: Vhori = set of horizontal lines Vvert = set of vertical lines

• |V | = n+n

Regularity Method for Hypergraphs An Extremal Problem

7

Solymosi: Ruzsa–Szemerédi ⇒ Ajtai–Szemerédi

• Let Z ⊆ [n]2 not containing a homothetic copy of an isosceles triangle

Consider the tripartite graph G = (V, E) with V = Vhori ∪ Vvert ∪ Vdiag where: Vhori = set of horizontal lines Vvert = set of vertical lines Vdiag = set of diagonal lines

• |V | = n+n+2n − 1 = O(n)

Regularity Method for Hypergraphs An Extremal Problem

7

Solymosi: Ruzsa–Szemerédi ⇒ Ajtai–Szemerédi

• Let Z ⊆ [n]2 not containing a homothetic copy of an isosceles triangle

Consider the tripartite graph G = (V, E) with V = Vhori ∪ Vvert ∪ Vdiag where: Vhori = set of horizontal lines Vvert = set of vertical lines Vdiag = set of diagonal lines and E =

z∈Z

• |V | = n+n+2n − 1 = O(n)

Regularity Method for Hypergraphs An Extremal Problem

7

Solymosi: Ruzsa–Szemerédi ⇒ Ajtai–Szemerédi

• Let Z ⊆ [n]2 not containing a homothetic copy of an isosceles triangle

Consider the tripartite graph G = (V, E) with V = Vhori ∪ Vvert ∪ Vdiag where: Vhori = set of horizontal lines Vvert = set of vertical lines Vdiag = set of diagonal lines and E =

z∈Z

• {h(z), v(z)},
• |V | = n+n+2n − 1 = O(n)

Regularity Method for Hypergraphs An Extremal Problem

7

Solymosi: Ruzsa–Szemerédi ⇒ Ajtai–Szemerédi

• Let Z ⊆ [n]2 not containing a homothetic copy of an isosceles triangle

Consider the tripartite graph G = (V, E) with V = Vhori ∪ Vvert ∪ Vdiag where: Vhori = set of horizontal lines Vvert = set of vertical lines Vdiag = set of diagonal lines and E =

z∈Z

• {h(z), v(z)},

{h(z), d(z)},

• |V | = n+n+2n − 1 = O(n)

Regularity Method for Hypergraphs An Extremal Problem

7

Solymosi: Ruzsa–Szemerédi ⇒ Ajtai–Szemerédi

• Let Z ⊆ [n]2 not containing a homothetic copy of an isosceles triangle

Consider the tripartite graph G = (V, E) with V = Vhori ∪ Vvert ∪ Vdiag where: Vhori = set of horizontal lines Vvert = set of vertical lines Vdiag = set of diagonal lines and E =

z∈Z

• {h(z), v(z)},

{h(z), d(z)},{v(z), d(z)}

• |V | = n+n+2n − 1 = O(n)

Regularity Method for Hypergraphs An Extremal Problem

7

Solymosi: Ruzsa–Szemerédi ⇒ Ajtai–Szemerédi

• Let Z ⊆ [n]2 not containing a homothetic copy of an isosceles triangle

Consider the tripartite graph G = (V, E) with V = Vhori ∪ Vvert ∪ Vdiag where: Vhori = set of horizontal lines Vvert = set of vertical lines Vdiag = set of diagonal lines and E =

z∈Z

• {h(z), v(z)},

{h(z), d(z)},{v(z), d(z)}

• |V | = n+n+2n − 1 = O(n)
• |E| = 3|Z| and every edge is in at least one triangle

Regularity Method for Hypergraphs An Extremal Problem

7

Solymosi: Ruzsa–Szemerédi ⇒ Ajtai–Szemerédi

• Let Z ⊆ [n]2 not containing a homothetic copy of an isosceles triangle

Consider the tripartite graph G = (V, E) with V = Vhori ∪ Vvert ∪ Vdiag where: Vhori = set of horizontal lines Vvert = set of vertical lines Vdiag = set of diagonal lines and E =

z∈Z

• {h(z), v(z)},

{h(z), d(z)},{v(z), d(z)}

• |V | = n+n+2n − 1 = O(n)
• |E| = 3|Z| and every edge is in at least one triangle
• assumption on Z ⇒ every edge is in at most one triangle

Regularity Method for Hypergraphs An Extremal Problem

7

Solymosi: Ruzsa–Szemerédi ⇒ Ajtai–Szemerédi

• Let Z ⊆ [n]2 not containing a homothetic copy of an isosceles triangle

Consider the tripartite graph G = (V, E) with V = Vhori ∪ Vvert ∪ Vdiag where: Vhori = set of horizontal lines Vvert = set of vertical lines Vdiag = set of diagonal lines and E =

z∈Z

• {h(z), v(z)},

{h(z), d(z)},{v(z), d(z)}

• |V | = n+n+2n − 1 = O(n)
• |E| = 3|Z| and every edge is in at least one triangle
• assumption on Z ⇒ every edge is in at most one triangle

⇒ 3|Z| = |E| = o(|V |2) = o(n2)

Regularity Method for Hypergraphs An Extremal Problem

8

Regularity Lemma for Graphs

Theorem (Szemerédi 1978). For every real ε > 0 and every integer t0 there exist some T0 such that for every graph G = (V, E) there exist a partition of V = V1 ∪ · · · ∪ Vt satisfying

Regularity Method for Hypergraphs An Extremal Problem

8

Regularity Lemma for Graphs

Theorem (Szemerédi 1978). For every real ε > 0 and every integer t0 there exist some T0 such that for every graph G = (V, E) there exist a partition of V = V1 ∪ · · · ∪ Vt satisfying (i) (boundedness) t0 ≤ t ≤ T0, (ii) (equitability) |V1| ≤ · · · ≤ |Vt| ≤ |V1| + 1

Regularity Method for Hypergraphs An Extremal Problem

8

Regularity Lemma for Graphs

Theorem (Szemerédi 1978). For every real ε > 0 and every integer t0 there exist some T0 such that for every graph G = (V, E) there exist a partition of V = V1 ∪ · · · ∪ Vt satisfying (i) (boundedness) t0 ≤ t ≤ T0, (ii) (equitability) |V1| ≤ · · · ≤ |Vt| ≤ |V1| + 1 (iii) (regularity) for all but εt2 pairs {i, j} ∈

[t]

2

• the induced subgraph

G[Vi, Vj] is ε-regular

Regularity Method for Hypergraphs An Extremal Problem

8

Regularity Lemma for Graphs

Theorem (Szemerédi 1978). For every real ε > 0 and every integer t0 there exist some T0 such that for every graph G = (V, E) there exist a partition of V = V1 ∪ · · · ∪ Vt satisfying (i) (boundedness) t0 ≤ t ≤ T0, (ii) (equitability) |V1| ≤ · · · ≤ |Vt| ≤ |V1| + 1 (iii) (regularity) for all but εt2 pairs {i, j} ∈

[t]

2

• the induced subgraph

G[Vi, Vj] is ε-regular, i.e., for all Ui ⊆ Vi and Uj ⊆ Vj with

Regularity Method for Hypergraphs An Extremal Problem

8

Regularity Lemma for Graphs

Theorem (Szemerédi 1978). For every real ε > 0 and every integer t0 there exist some T0 such that for every graph G = (V, E) there exist a partition of V = V1 ∪ · · · ∪ Vt satisfying (i) (boundedness) t0 ≤ t ≤ T0, (ii) (equitability) |V1| ≤ · · · ≤ |Vt| ≤ |V1| + 1 (iii) (regularity) for all but εt2 pairs {i, j} ∈

[t]

2

• the induced subgraph

G[Vi, Vj] is ε-regular, i.e., for all Ui ⊆ Vi and Uj ⊆ Vj with |Ui| ≥ ε|Vi| and |Uj| ≥ ε|Vj| we have

Regularity Method for Hypergraphs An Extremal Problem

8

Regularity Lemma for Graphs

Theorem (Szemerédi 1978). For every real ε > 0 and every integer t0 there exist some T0 such that for every graph G = (V, E) there exist a partition of V = V1 ∪ · · · ∪ Vt satisfying (i) (boundedness) t0 ≤ t ≤ T0, (ii) (equitability) |V1| ≤ · · · ≤ |Vt| ≤ |V1| + 1 (iii) (regularity) for all but εt2 pairs {i, j} ∈

[t]

2

• the induced subgraph

G[Vi, Vj] is ε-regular, i.e., for all Ui ⊆ Vi and Uj ⊆ Vj with |Ui| ≥ ε|Vi| and |Uj| ≥ ε|Vj| we have

• |G[Ui, Uj]|

|Ui||Uj| − |G[Vi, Vj]| |Vi||Vj|

• < ε .

Regularity Method for Hypergraphs An Extremal Problem

9

Counting Lemma for Graphs

• Fact. Let 1 ≥ d ≥ 2ε > 0.

Short version.

Regularity Method for Hypergraphs An Extremal Problem

9

Counting Lemma for Graphs

• Fact. Let 1 ≥ d ≥ 2ε > 0. If
• G = (V1∪V2∪V3, E) is a tripartite graph with |V1| = |V2| = |V3| = m

Short version. tripartite,

Regularity Method for Hypergraphs An Extremal Problem

9

Counting Lemma for Graphs

• Fact. Let 1 ≥ d ≥ 2ε > 0. If
• G = (V1∪V2∪V3, E) is a tripartite graph with |V1| = |V2| = |V3| = m

and for all 1 ≤ i < j ≤ 3

• G[Vi, Vj] is ε-regular

Short version. tripartite, ε-regular,

Regularity Method for Hypergraphs An Extremal Problem

9

Counting Lemma for Graphs

• Fact. Let 1 ≥ d ≥ 2ε > 0. If
• G = (V1∪V2∪V3, E) is a tripartite graph with |V1| = |V2| = |V3| = m

and for all 1 ≤ i < j ≤ 3

• G[Vi, Vj] is ε-regular and
• G[Vi, Vj]/m2 ≥ d,

Short version. tripartite, ε-regular, and d-dense

Regularity Method for Hypergraphs An Extremal Problem

9

Counting Lemma for Graphs

• Fact. Let 1 ≥ d ≥ 2ε > 0. If
• G = (V1∪V2∪V3, E) is a tripartite graph with |V1| = |V2| = |V3| = m

and for all 1 ≤ i < j ≤ 3

• G[Vi, Vj] is ε-regular and
• G[Vi, Vj]/m2 ≥ d,

then Short version. tripartite, ε-regular, and d-dense ⇒

Regularity Method for Hypergraphs An Extremal Problem

9

Counting Lemma for Graphs

• Fact. Let 1 ≥ d ≥ 2ε > 0. If
• G = (V1∪V2∪V3, E) is a tripartite graph with |V1| = |V2| = |V3| = m

and for all 1 ≤ i < j ≤ 3

• G[Vi, Vj] is ε-regular and
• G[Vi, Vj]/m2 ≥ d,

then G contains at least (1 − 2ε)(d − ε)3m3 triangles. Short version. tripartite, ε-regular, and d-dense ⇒ ∼ d3m3 triangles

Regularity Method for Hypergraphs An Extremal Problem

10

The Regularity Method

Many more applications of the Regularity Lemma, the Counting Lemma and its extensions in: Extremal Graph Theory Ramsey–Turán problems, (6, 3)-problem, (weak) Burr–Erd˝

• s Conjecture, Pósa–Seymour Conjecture, Alon–Yuster Con-

jecture

Regularity Method for Hypergraphs An Extremal Problem

10

The Regularity Method

Many more applications of the Regularity Lemma, the Counting Lemma and its extensions in: Extremal Graph Theory Ramsey–Turán problems, (6, 3)-problem, (weak) Burr–Erd˝

• s Conjecture, Pósa–Seymour Conjecture, Alon–Yuster Con-

jecture Number Theory & Discrete Geometry r3 = o(n), Solymosi’s proof of the Ajtai–Szemerédi theorem, Balog–Szemerédi Theorem Theoretical Computer Science Algorithmic versions, Network designs, Property testing, approximations of NP-hard problems, e.g., Max-Cut

Regularity Method for Hypergraphs An Extremal Problem

10

The Regularity Method

Many more applications of the Regularity Lemma, the Counting Lemma and its extensions in: Extremal Graph Theory Ramsey–Turán problems, (6, 3)-problem, (weak) Burr–Erd˝

• s Conjecture, Pósa–Seymour Conjecture, Alon–Yuster Con-

jecture Number Theory & Discrete Geometry r3 = o(n), Solymosi’s proof of the Ajtai–Szemerédi theorem, Balog–Szemerédi Theorem Theoretical Computer Science Algorithmic versions, Network designs, Property testing, approximations of NP-hard problems, e.g., Max-Cut

• Hope. Extension to hypergraphs is useful in the same areas.

Regularity Method for Hypergraphs

11

Review/Outline

1 Density Theorems Szemerédi’s Density Theorem Density Theorems of Furstenberg and Katznelson 2 An Extremal Hypergraph problem Connection to the Density Theorems Szemeredi’s Regularity Lemma for Graphs Solution of the Extremal Problem for Graphs

Regularity Method for Hypergraphs

11

Review/Outline

1 Density Theorems Szemerédi’s Density Theorem Density Theorems of Furstenberg and Katznelson 2 An Extremal Hypergraph problem Connection to the Density Theorems Szemeredi’s Regularity Lemma for Graphs Solution of the Extremal Problem for Graphs 3 Regularity Method for Hypergraphs History Regularity Lemma Counting Lemma

Regularity Method for Hypergraphs

12

History

Bad News. Straightforward approach fails.

Regularity Method for Hypergraphs

12

History

Bad News. Straightforward approach fails. 92–02 Frankl & Rödl: Regularity Lemma for k = 3 and corresponding Counting Lemma for K(3)

4

Regularity Method for Hypergraphs

12

History

Bad News. Straightforward approach fails. 92–02 Frankl & Rödl: Regularity Lemma for k = 3 and corresponding Counting Lemma for K(3)

4

− → solution of the extremal problem for k = 3

Regularity Method for Hypergraphs

12

History

Bad News. Straightforward approach fails. 92–02 Frankl & Rödl: Regularity Lemma for k = 3 and corresponding Counting Lemma for K(3)

4

− → solution of the extremal problem for k = 3 99–03 Nagle & Rödl: Counting Lemma for arbitrary fixed 3-graphs

Regularity Method for Hypergraphs

12

History

Bad News. Straightforward approach fails. 92–02 Frankl & Rödl: Regularity Lemma for k = 3 and corresponding Counting Lemma for K(3)

4

− → solution of the extremal problem for k = 3 99–03 Nagle & Rödl: Counting Lemma for arbitrary fixed 3-graphs 00–04 Rödl & Skokan: general Hypergraph Regularity Lemma k ≥ 3 and Counting Lemma for K(4)

5

Regularity Method for Hypergraphs

12

History

Bad News. Straightforward approach fails. 92–02 Frankl & Rödl: Regularity Lemma for k = 3 and corresponding Counting Lemma for K(3)

4

− → solution of the extremal problem for k = 3 99–03 Nagle & Rödl: Counting Lemma for arbitrary fixed 3-graphs 00–04 Rödl & Skokan: general Hypergraph Regularity Lemma k ≥ 3 and Counting Lemma for K(4)

5

− → solution of the extremal problem for k = 4

Regularity Method for Hypergraphs

12

History

Bad News. Straightforward approach fails. 92–02 Frankl & Rödl: Regularity Lemma for k = 3 and corresponding Counting Lemma for K(3)

4

− → solution of the extremal problem for k = 3 99–03 Nagle & Rödl: Counting Lemma for arbitrary fixed 3-graphs 00–04 Rödl & Skokan: general Hypergraph Regularity Lemma k ≥ 3 and Counting Lemma for K(4)

5

− → solution of the extremal problem for k = 4 03–04 Gowers: different Hypergraph Regularity Lemma and a corresponding Counting Lemma

Regularity Method for Hypergraphs

12

History

Bad News. Straightforward approach fails. 92–02 Frankl & Rödl: Regularity Lemma for k = 3 and corresponding Counting Lemma for K(3)

4

− → solution of the extremal problem for k = 3 99–03 Nagle & Rödl: Counting Lemma for arbitrary fixed 3-graphs 00–04 Rödl & Skokan: general Hypergraph Regularity Lemma k ≥ 3 and Counting Lemma for K(4)

5

− → solution of the extremal problem for k = 4 03–04 Gowers: different Hypergraph Regularity Lemma and a corresponding Counting Lemma − → solution to the extremal problem for arbitrary k

Regularity Method for Hypergraphs

12

History

Bad News. Straightforward approach fails. 92–02 Frankl & Rödl: Regularity Lemma for k = 3 and corresponding Counting Lemma for K(3)

4

− → solution of the extremal problem for k = 3 99–03 Nagle & Rödl: Counting Lemma for arbitrary fixed 3-graphs 00–04 Rödl & Skokan: general Hypergraph Regularity Lemma k ≥ 3 and Counting Lemma for K(4)

5

− → solution of the extremal problem for k = 4 03–04 Gowers: different Hypergraph Regularity Lemma and a corresponding Counting Lemma − → solution to the extremal problem for arbitrary k 03–04 Nagle, Rödl & S.: general Counting Lemma corresponding to the Rödl– Skokan Lemma

Regularity Method for Hypergraphs

12

History

Bad News. Straightforward approach fails. 92–02 Frankl & Rödl: Regularity Lemma for k = 3 and corresponding Counting Lemma for K(3)

4

− → solution of the extremal problem for k = 3 99–03 Nagle & Rödl: Counting Lemma for arbitrary fixed 3-graphs 00–04 Rödl & Skokan: general Hypergraph Regularity Lemma k ≥ 3 and Counting Lemma for K(4)

5

− → solution of the extremal problem for k = 4 03–04 Gowers: different Hypergraph Regularity Lemma and a corresponding Counting Lemma − → solution to the extremal problem for arbitrary k 03–04 Nagle, Rödl & S.: general Counting Lemma corresponding to the Rödl– Skokan Lemma − → solution to the extremal problem for arbitrary k

Regularity Method for Hypergraphs

13

The Regularity Lemma (k = 3)

Theorem (Frankl & Rödl 2002). For all reals µ > 0 and δ3 > 0 and all functions δ2: N → (0, 1] and r: N2 → N there are integers T0 and n0 such that the following holds.

Regularity Method for Hypergraphs

13

The Regularity Lemma (k = 3)

Theorem (Frankl & Rödl 2002). For all reals µ > 0 and δ3 > 0 and all functions δ2: N → (0, 1] and r: N2 → N there are integers T0 and n0 such that the following holds. For any 3-uniform hypergraph H(3) on n ≥ n0 vertices

Regularity Method for Hypergraphs

13

The Regularity Lemma (k = 3)

Theorem (Frankl & Rödl 2002). For all reals µ > 0 and δ3 > 0 and all functions δ2: N → (0, 1] and r: N2 → N there are integers T0 and n0 such that the following holds. For any 3-uniform hypergraph H(3) on n ≥ n0 vertices there exist a partition P = {P(1), P(2)} and integers t and ℓ such that

Regularity Method for Hypergraphs

13

The Regularity Lemma (k = 3)

Theorem (Frankl & Rödl 2002). For all reals µ > 0 and δ3 > 0 and all functions δ2: N → (0, 1] and r: N2 → N there are integers T0 and n0 such that the following holds. For any 3-uniform hypergraph H(3) on n ≥ n0 vertices there exist a partition P = {P(1), P(2)} and integers t and ℓ such that (i) P is T0-bounded, i.e, max{t, ℓ} < T0,

Regularity Method for Hypergraphs

13

The Regularity Lemma (k = 3)

Theorem (Frankl & Rödl 2002). For all reals µ > 0 and δ3 > 0 and all functions δ2: N → (0, 1] and r: N2 → N there are integers T0 and n0 such that the following holds. For any 3-uniform hypergraph H(3) on n ≥ n0 vertices there exist a partition P = {P(1), P(2)} and integers t and ℓ such that (i) P is T0-bounded, i.e, max{t, ℓ} < T0, (ii) P is (µ, δ2(ℓ), 1/ℓ = d2)-equitable

Regularity Method for Hypergraphs

13

The Regularity Lemma (k = 3)

Theorem (Frankl & Rödl 2002). For all reals µ > 0 and δ3 > 0 and all functions δ2: N → (0, 1] and r: N2 → N there are integers T0 and n0 such that the following holds. For any 3-uniform hypergraph H(3) on n ≥ n0 vertices there exist a partition P = {P(1), P(2)} and integers t and ℓ such that (i) P is T0-bounded, i.e, max{t, ℓ} < T0, (ii) P is (µ, δ2(ℓ), 1/ℓ = d2)-equitable (iii) H(3) is (δ3, r(t, ℓ))-regular w.r.t. P.

Regularity Method for Hypergraphs

14

The Counting Lemma (k = 3)

Theorem (Frankl & Rödl 2002). ∀ γ > 0, d3 > 0

Regularity Method for Hypergraphs

14

The Counting Lemma (k = 3)

Theorem (Frankl & Rödl 2002). ∀ γ > 0, d3 > 0 ∃ δ3 > 0:

Regularity Method for Hypergraphs

14

The Counting Lemma (k = 3)

Theorem (Frankl & Rödl 2002). ∀ γ > 0, d3 > 0 ∃ δ3 > 0: ∀ d2 > 0

Regularity Method for Hypergraphs

14

The Counting Lemma (k = 3)

Theorem (Frankl & Rödl 2002). ∀ γ > 0, d3 > 0 ∃ δ3 > 0: ∀ d2 > 0 ∃ δ2, r, m0 such that

Regularity Method for Hypergraphs

14

The Counting Lemma (k = 3)

Theorem (Frankl & Rödl 2002). ∀ γ > 0, d3 > 0 ∃ δ3 > 0: ∀ d2 > 0 ∃ δ2, r, m0 such that every ((δ3, δ2), (d3, d2), r)-regular

Regularity Method for Hypergraphs

14

The Counting Lemma (k = 3)

Theorem (Frankl & Rödl 2002). ∀ γ > 0, d3 > 0 ∃ δ3 > 0: ∀ d2 > 0 ∃ δ2, r, m0 such that every ((δ3, δ2), (d3, d2), r)-regular (m, 4, 3)-complex

Regularity Method for Hypergraphs

14

The Counting Lemma (k = 3)

Theorem (Frankl & Rödl 2002). ∀ γ > 0, d3 > 0 ∃ δ3 > 0: ∀ d2 > 0 ∃ δ2, r, m0 such that every ((δ3, δ2), (d3, d2), r)-regular (m, 4, 3)-complex

H = {H(1), H(2), H(3)}

Regularity Method for Hypergraphs

14

The Counting Lemma (k = 3)

Theorem (Frankl & Rödl 2002). ∀ γ > 0, d3 > 0 ∃ δ3 > 0: ∀ d2 > 0 ∃ δ2, r, m0 such that every ((δ3, δ2), (d3, d2), r)-regular (m, 4, 3)-complex

H = {H(1), H(2), H(3)}

with m ≥ m0, contains (1 ± γ)d6

2d4 3m4

copies of K(3)

4

.

Regularity Method for Hypergraphs

15

Open Problems

• Is there a proof of the density version of the Hales–Jewett Theorem

based on the extremal hypergraph problem

Regularity Method for Hypergraphs

15

Open Problems

• Is there a proof of the density version of the Hales–Jewett Theorem

based on the extremal hypergraph problem (or based on the Regularity Method for hypergraphs)?

Regularity Method for Hypergraphs

15

Open Problems

• Is there a proof of the density version of the Hales–Jewett Theorem

based on the extremal hypergraph problem (or based on the Regularity Method for hypergraphs)?

• What about “Blow-up type” extensions of the Counting Lemma?