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Mathias Schacht

November 2002

Outline

1 Introduction

Extremal Graph Theory Random Graph Theory

Extremal Random Graph Theory The Main Result

1 Introduction

Extremal Graph Theory Random Graph Theory

Extremal Random Graph Theory The Main Result

2 Proof of the Main Result

Szemer ´edi’s regularity lemma for sparse graphs

The Counting Lemma for subgraphs for random graphs Outline of the proof the Main Result

Outline

1 Introduction

Extremal Graph Theory Random Graph Theory

Extremal Random Graph Theory The Main Result

2 Proof of the Main Result

Szemer ´edi’s regularity lemma for sparse graphs

The Counting Lemma for subgraphs for random graphs Outline of the proof the Main Result

3 Proof of the Counting Lemma

Outline of the proof of the Counting Lemma for H = K₄ − e The k-tuple Lemma

The “Pick-Up” Lemma

Extremal (Deterministic) Graph Theory

Motivation.

What is the structure of a H-free graph with a maximum number of edges, for some fixed graph H?

Extremal (Deterministic) Graph Theory

Motivation.

What is the structure of a H-free graph with a maximum number of edges, for some fixed graph H?

Which properties imply the existence of a certain fixed subgraph H?

Definition. ex(G, H) = max{e(F)|H 6⊆ F ⊆ G}

Results in Extremal Graph Theory

Definition. ex(G, H) = max{e(F)|H 6⊆ F ⊆ G} Observation. For every graph G and H

ex(G, H) ≥ e(G) − #{H⁰ ⊆ G} for every H⁰ ⊆ H.

Definition. ex(G, H) = max{e(F)|H 6⊆ F ⊆ G} Observation. For every graph G and H

ex(G, H) ≥ e(G) − #{H⁰ ⊆ G} for every H⁰ ⊆ H.

Theorem (Tur ´an 1941).

ex(K_n, K_l) =

1 − 1

l − 1 + O

1 n

n 2

Results in Extremal Graph Theory

Definition. ex(G, H) = max{e(F)|H 6⊆ F ⊆ G} Observation. For every graph G and H

ex(G, H) ≥ e(G) − #{H⁰ ⊆ G} for every H⁰ ⊆ H.

Theorem (Tur ´an 1941).

ex(K_n, K_l) =

1 − 1

l − 1 + O

1 n

n 2

Theorem (Erd ˝os–Stone–Simonovits 1946, 1966). For every graph H ex(K_n, H) = 1 − 1

χ(H) − 1 + o(1)

! n

2

.

Definition (G(n, p)).

For 0 ≤ p = p(n) ≤ 1, the binomial random graph G in G(n, p) has n vertices and each edge occurs independently with probability p.

Random Graph Theory

Definition (G(n, p)).

For 0 ≤ p = p(n) ≤ 1, the binomial random graph G in G(n, p) has n vertices and each edge occurs independently with probability p.

Equivalently, G(n, p) is the probabilty space over all labeled graphs on n vertices by setting

P_{(G) = p}^e(G)_{(1 − p)(}

n

2)₋^e(G).

Definition (G(n, p)).

For 0 ≤ p = p(n) ≤ 1, the binomial random graph G in G(n, p) has n vertices and each edge occurs independently with probability p.

Equivalently, G(n, p) is the probabilty space over all labeled graphs on n vertices by setting

P_{(G) = p}^e(G)_{(1 − p)(}

n

2)₋^e(G).

We say a property P holds asymptotically almost surely (a.a.s.) if P_{(G ∈ G(n, p) satisfies P) = 1 − o(1).}

Extremal Random Graph Theory

For 0 ≤ p = p(n) ≤ 1 consider the random variable:

ex(G(n, p), H)

For 0 ≤ p = p(n) ≤ 1 consider the random variable:

ex(G(n, p), H)

Question. For which p does

ex(G(n, p), H) = 1 + 1

χ(H) − 1 + o(1)

!

pn 2

(1) hold a.a.s.?

Extremal Random Graph Theory

For 0 ≤ p = p(n) ≤ 1 consider the random variable:

ex(G(n, p), H)

Question. For which p does

ex(G(n, p), H) = 1 + 1

χ(H) − 1 + o(1)

!

pn 2

(1) hold a.a.s.?

Fact. If (1) holds a.a.s. for some p₁, then it does a.a.s. for each p₂ ≥ p₁. Question. What is the minimal p = p(n)? What is the threshold?

Main Conjecture

Definition (2-density). Let H be a graph with v(H) ≥ 3 m₂(H) = max

(e(H⁰) − 1 v(H⁰) − 2

H⁰ ⊆ H,|V (H⁰)| ≥ 3

)

.

Main Conjecture

Definition (2-density). Let H be a graph with v(H) ≥ 3 m₂(H) = max

(e(H⁰) − 1 v(H⁰) − 2

H⁰ ⊆ H,|V (H⁰)| ≥ 3

)

.

Conjecture (Kohayakawa, Łuczak & R ¨odl 1997). Let H be a non-empty graph with v(H) ≥ 3 and let 0 ≤ p(n) ≤ 1 such that p(n) n^−1/m2(H). Then a.a.s.

ex(G, H) = 1 + 1

χ(H) − 1 + o(1)

!

e(G) holds for G in G(n, p).

The Lower Bound ( p ( n ) n

)

Recall. For every graph G and H

ex(G, H) ≥ e(G) − #{H⁰ ⊆ G} for every H⁰ ⊆ H.

The Lower Bound ( p ( n ) n

)

Recall. For every graph G and H

ex(G, H) ≥ e(G) − #{H⁰ ⊆ G} for every H⁰ ⊆ H.

Observation.

p(n) n^−1/m2(H) ⇔ p(n)n² p(n)^e(H0)n^v(H0) for every H⁰ ⊆ H.

Definition (d-degenerate). A graph H is d-degenerate, if d(H) = max

H⁰⊆H δ(H⁰) ≤ d.

Main Result

Definition (d-degenerate). A graph H is d-degenerate, if d(H) = max

H⁰⊆H δ(H⁰) ≤ d.

Theorem (Main Result). For every graph H, real δ > 0, and p = p(n) (logn)⁴/n^1/d(H),

a graph G in G(n, p) a.a.s. satisfies the following property:

If F is an arbitrary, not necessarily induced subgraph of G with e(F) ≥ 1 − 1

χ(H) − 1 + δ

!

pn 2

, then H ⊆ F.

Corollary (H = K_l). For

p(n) (logn)⁴ n

!1/(l−1)

G in G(n, p) a.a.s. satisfies

ex(G, K_l) =

1 + 1

l + o(1)

e(G).

Corollary (H = K_l). For

p(n) (logn)⁴ n

!1/(l−1)

G in G(n, p) a.a.s. satisfies

ex(G, K_l) =

1 + 1

l + o(1)

e(G).

Remark. KŁR–Conjecture for H = K_l: p(n)

1 n

_2/(l+1)

is sufficient.

Related Work

H is a forest follows from a simple application of Szemer ´edi’s regu- larity lemma for sparse graphs

Related Work

H is a forest follows from a simple application of Szemer ´edi’s regu- larity lemma for sparse graphs

H = K₃ follows from a result of Frankl and R ¨odl, 1986 H = C₄ follows from a result of F ¨uredi 1994

Related Work

H is a forest follows from a simple application of Szemer ´edi’s regu- larity lemma for sparse graphs

H = K₃ follows from a result of Frankl and R ¨odl, 1986 H = C₄ follows from a result of F ¨uredi 1994

H = C_l proved by Haxell, Kohayakawa & Łuczak, 1995 and 1996

Related Work

H is a forest follows from a simple application of Szemer ´edi’s regu- larity lemma for sparse graphs

H = K₃ follows from a result of Frankl and R ¨odl, 1986 H = C₄ follows from a result of F ¨uredi 1994

H = C_l proved by Haxell, Kohayakawa & Łuczak, 1995 and 1996

H = K₄ proved by Kohayakawa, Łuczak & R ¨odl, 1997

H = K₅ follows from a stronger result proved by Gerke, Schickinger & Steger, 2002+

Related Work

H is a forest follows from a simple application of Szemer ´edi’s regu- larity lemma for sparse graphs

H = K₃ follows from a result of Frankl and R ¨odl, 1986 H = C₄ follows from a result of F ¨uredi 1994

H = C_l proved by Haxell, Kohayakawa & Łuczak, 1995 and 1996

H = K₄ proved by Kohayakawa, Łuczak & R ¨odl, 1997

H = K₅ follows from a stronger result proved by Gerke, Schickinger & Steger, 2002+

H = K_l Szab ´o and Vu proved it for p(n) n^{−1/(l−1.5)}, 2002+

Summary & Outline

1 Introduction

Extremal Graph Theory Random Graph Theory

Extremal Random Graph Theory The Main Result

1 Introduction

Extremal Graph Theory Random Graph Theory

Extremal Random Graph Theory The Main Result

2 Proof of the Main Result

Szemer ´edi’s regularity lemma for sparse graphs

The Counting Lemma for subgraphs for random graphs Outline of the proof the Main Result

3 Proof of the Counting Lemma

Outline of the proof of the Counting Lemma for H = K₄ − e The k-tuple Lemma

The “Pick-Up” Lemma

Preliminary Definitions

Definition ((ε, p)-regular). For G = (V, E), ε > 0, U, W ⊆ V disjoint, we say (U,W) is (ε, p)-regular if for all U⁰ ⊆ U, W⁰ ⊆ W with |U⁰| ≥ ε|U| and |W⁰| ≥ ε|W| we have

e_G(U⁰, W⁰)

p|U⁰||W⁰| − e_G(U, W) p|U||W|

≤ ε.

Definition ((ε, p)-regular). For G = (V, E), ε > 0, U, W ⊆ V disjoint, we say (U,W) is (ε, p)-regular if for all U⁰ ⊆ U, W⁰ ⊆ W with |U⁰| ≥ ε|U| and |W⁰| ≥ ε|W| we have

e_G(U⁰, W⁰)

p|U⁰||W⁰| − e_G(U, W) p|U||W|

≤ ε.

Definition ((ξ, C)-bounded). For ξ > 0, C > 1, and 0 ≤ p ≤ 1 we say that G = (V, E) is (ξ, C)-bounded with respect to density p, if for all U⁰, W⁰ ⊆ V not necessarily disjoint with |U⁰|,|V ⁰| ≥ ξ|V | we have

e_G(U⁰, W⁰) ≤ Cp |U⁰||W⁰| − |U⁰ ∩ W⁰| + 1 2

!

.

Szemer ´edi’s regularity lemma for sparse graphs

Theorem (Kohayakawa, R ¨odl 1997). For every ε > 0, C > 1, and t₀ ≥ 1, there exist constants ξ = ξ(ε, C, t₀) and T₀ = T₀(ε, C, t₀) such that the vertex set of any graph G = (V, E) that is (ξ, C)-bounded with respect to density p can be partitioned V = V₀ ∪ V₁ ∪ · · · ∪ V_t such that

Theorem (Kohayakawa, R ¨odl 1997). For every ε > 0, C > 1, and t₀ ≥ 1, there exist constants ξ = ξ(ε, C, t₀) and T₀ = T₀(ε, C, t₀) such that the vertex set of any graph G = (V, E) that is (ξ, C)-bounded with respect to density p can be partitioned V = V₀ ∪ V₁ ∪ · · · ∪ V_t such that

(i) t₀ ≤ t ≤ T₀

Szemer ´edi’s regularity lemma for sparse graphs

Theorem (Kohayakawa, R ¨odl 1997). For every ε > 0, C > 1, and t₀ ≥ 1, there exist constants ξ = ξ(ε, C, t₀) and T₀ = T₀(ε, C, t₀) such that the vertex set of any graph G = (V, E) that is (ξ, C)-bounded with respect to density p can be partitioned V = V₀ ∪ V₁ ∪ · · · ∪ V_t such that

(i) t₀ ≤ t ≤ T₀

(ii) V₀ ≤ ε|V | and |V₁| = |V₂| = · · · = |V_t|

Theorem (Kohayakawa, R ¨odl 1997). For every ε > 0, C > 1, and t₀ ≥ 1, there exist constants ξ = ξ(ε, C, t₀) and T₀ = T₀(ε, C, t₀) such that the vertex set of any graph G = (V, E) that is (ξ, C)-bounded with respect to density p can be partitioned V = V₀ ∪ V₁ ∪ · · · ∪ V_t such that

(i) t₀ ≤ t ≤ T₀

(ii) V₀ ≤ ε|V | and |V₁| = |V₂| = · · · = |V_t|

(iii) all but at most ε₂^t pairs (V_i, V_j) are (ε, p)-regular

Szemer ´edi’s regularity lemma for sparse graphs

Theorem (Kohayakawa, R ¨odl 1997). For every ε > 0, C > 1, and t₀ ≥ 1, there exist constants ξ = ξ(ε, C, t₀) and T₀ = T₀(ε, C, t₀) such that the vertex set of any graph G = (V, E) that is (ξ, C)-bounded with respect to density p can be partitioned V = V₀ ∪ V₁ ∪ · · · ∪ V_t such that

(i) t₀ ≤ t ≤ T₀

(ii) V₀ ≤ ε|V | and |V₁| = |V₂| = · · · = |V_t|

(iii) all but at most ε₂^t pairs (V_i, V_j) are (ε, p)-regular

A partition V = V₀ ∪ V₁ ∪ · · · ∪ V_t satisfying (i)–(iii) we call (ε, t)-regular partition.

• H = (V, E) is a fixed d-degenerate graph on h

• order V = {w₁, . . . , w_h} such that |Γ(w_i) ∩ {w₁, . . . , w_i−1}| ≤ d

Set-up

• H = (V, E) is a fixed d-degenerate graph on h

• order V = {w₁, . . . , w_h} such that |Γ(w_i) ∩ {w₁, . . . , w_i−1}| ≤ d

• t ≥ h is a fixed integer and n is sufficiently large

• 0 < α, ε < 1 are real constants

• H = (V, E) is a fixed d-degenerate graph on h

• order V = {w₁, . . . , w_h} such that |Γ(w_i) ∩ {w₁, . . . , w_i−1}| ≤ d

• t ≥ h is a fixed integer and n is sufficiently large

• 0 < α, ε < 1 are real constants

• G is a graph in G(n, p)

• J ⊆ G is h-partite with V (J) = V₁ ∪ · · · ∪ V_h, denote J[V_i, V_j] by J_ij

Set-up

• H = (V, E) is a fixed d-degenerate graph on h

• order V = {w₁, . . . , w_h} such that |Γ(w_i) ∩ {w₁, . . . , w_i−1}| ≤ d

• t ≥ h is a fixed integer and n is sufficiently large

• 0 < α, ε < 1 are real constants

• G is a graph in G(n, p)

• J ⊆ G is h-partite with V (J) = V₁ ∪ · · · ∪ V_h, denote J[V_i, V_j] by J_ij Moreover, consider the following assertions.

(I) |V_i| = m = n/t (II) p^dn (logn)⁴

• H = (V, E) is a fixed d-degenerate graph on h

• order V = {w₁, . . . , w_h} such that |Γ(w_i) ∩ {w₁, . . . , w_i−1}| ≤ d

• t ≥ h is a fixed integer and n is sufficiently large

• 0 < α, ε < 1 are real constants

• G is a graph in G(n, p)

• J ⊆ G is h-partite with V (J) = V₁ ∪ · · · ∪ V_h, denote J[V_i, V_j] by J_ij Moreover, consider the following assertions.

(I) |V_i| = m = n/t (II) p^dn (logn)⁴

(III) for all 1 ≤ i < j ≤ h e(J_ij) =







T = αpm² {w_i, w_j} ∈ E(H) 0 {w_i, w_j} 6∈ E(H) (IV) J_ij is (ε, p)-regular.

The Counting Lemma for subgraphs of random graphs

Lemma (Counting Lemma). For all reals α, σ > 0, positive integer d and every d-degenerate graph H on h vertices, there exists a real ε > 0 such that for every fixed integer t ≥ h a random graph G in G(n, p) satisfies the following property a.a.s.:

Every subgraph J ⊆ G satisfying conditions (I)–(IV) contains at least (1 − σ)(αp)^e(H)m^h

copies of H.

Outline of the Proof of the Main Result

Facts.

(P1) For G in G(n, p) a.a.s. e(G) = (1 + o(1))pⁿ₂.

Outline of the Proof of the Main Result

Facts.

(P1) For G in G(n, p) a.a.s. e(G) = (1 + o(1))pⁿ₂.

(P2) ∀ ξ > 0, C > 1 if p = p(n) 1/n a.a.s. G in G(n, p) is (ξ, C)- bounded with respect to density p.

Outline of the Proof of the Main Result

Facts.

(P1) For G in G(n, p) a.a.s. e(G) = (1 + o(1))pⁿ₂.

(P2) ∀ ξ > 0, C > 1 if p = p(n) 1/n a.a.s. G in G(n, p) is (ξ, C)- bounded with respect to density p.

Sketch of the proof.

• typical “regularity lemma” proof

Outline of the Proof of the Main Result

Facts.

(P1) For G in G(n, p) a.a.s. e(G) = (1 + o(1))pⁿ₂.

(P2) ∀ ξ > 0, C > 1 if p = p(n) 1/n a.a.s. G in G(n, p) is (ξ, C)- bounded with respect to density p.

Sketch of the proof.

• typical “regularity lemma” proof

• Let H be the fixed d-degenerate graph.

Outline of the Proof of the Main Result

Facts.

(P1) For G in G(n, p) a.a.s. e(G) = (1 + o(1))pⁿ₂.

(P2) ∀ ξ > 0, C > 1 if p = p(n) 1/n a.a.s. G in G(n, p) is (ξ, C)- bounded with respect to density p.

Sketch of the proof.

• typical “regularity lemma” proof

• Let H be the fixed d-degenerate graph.

• Let G be a random graph in G(n, p) satisfying the Facts, and δ > 0.

Outline of the Proof of the Main Result

Facts.

(P1) For G in G(n, p) a.a.s. e(G) = (1 + o(1))pⁿ₂.

(P2) ∀ ξ > 0, C > 1 if p = p(n) 1/n a.a.s. G in G(n, p) is (ξ, C)- bounded with respect to density p.

Sketch of the proof.

• typical “regularity lemma” proof

• Let H be the fixed d-degenerate graph.

• Let G be a random graph in G(n, p) satisfying the Facts, and δ > 0.

• Let F ⊆ G with more than (1 − 1/(χ(H) − 1) + δ)pⁿ₂ edges.

Outline of the Proof of the Main Result

Facts.

(P1) For G in G(n, p) a.a.s. e(G) = (1 + o(1))pⁿ₂.

(P2) ∀ ξ > 0, C > 1 if p = p(n) 1/n a.a.s. G in G(n, p) is (ξ, C)- bounded with respect to density p.

Sketch of the proof.

• typical “regularity lemma” proof

• Let H be the fixed d-degenerate graph.

• Let G be a random graph in G(n, p) satisfying the Facts, and δ > 0.

• Let F ⊆ G with more than (1 − 1/(χ(H) − 1) + δ)pⁿ₂ edges.

• Choose α and σ appropriately and the Counting lemma yields ε^CL.

Outline of the Proof of the Main Result

Facts.

(P1) For G in G(n, p) a.a.s. e(G) = (1 + o(1))pⁿ₂.

(P2) ∀ ξ > 0, C > 1 if p = p(n) 1/n a.a.s. G in G(n, p) is (ξ, C)- bounded with respect to density p.

Sketch of the proof.

• typical “regularity lemma” proof

• Let H be the fixed d-degenerate graph.

• Let G be a random graph in G(n, p) satisfying the Facts, and δ > 0.

• Let F ⊆ G with more than (1 − 1/(χ(H) − 1) + δ)pⁿ₂ edges.

• Choose α and σ appropriately and the Counting lemma yields ε^CL.

• Obtain an (ε, t)-regular partition P of F and ξ for appropriate chosen C, ε = ε(ε^CL) and t₀ ≤ t by the regularity lemma for sparse graphs.

• Consider the cluster graph F_Cluster of F with respect to P, depending on ε and α.

Outline of the Proof of the Main Result (continued)

• Consider the cluster graph F_Cluster of F with respect to P, depending on ε and α.

• Use (P2) and “smart” choice of constants to show that e(F_Cluster) ≥ 1 − 1

χ(H) − 1 + δ⁰

! t

2

.

• Consider the cluster graph F_Cluster of F with respect to P, depending on ε and α.

• Use (P2) and “smart” choice of constants to show that e(F_Cluster) ≥ 1 − 1

χ(H) − 1 + δ⁰

! t

2

.

⇒ H ⊆ F_Cluster by Erd ˝os–Stone–Simonovits

Outline of the Proof of the Main Result (continued)

• Consider the cluster graph F_Cluster of F with respect to P, depending on ε and α.

• Use (P2) and “smart” choice of constants to show that e(F_Cluster) ≥ 1 − 1

χ(H) − 1 + δ⁰

! t

2

.

⇒ H ⊆ F_Cluster by Erd ˝os–Stone–Simonovits

• This copy of H translates to a J⁰ ⊆ F which almost satisfies the as- sumptions (set-up) of the Counting Lemma.

• Consider the cluster graph F_Cluster of F with respect to P, depending on ε and α.

• Use (P2) and “smart” choice of constants to show that e(F_Cluster) ≥ 1 − 1

χ(H) − 1 + δ⁰

! t

2

.

⇒ H ⊆ F_Cluster by Erd ˝os–Stone–Simonovits

• This copy of H translates to a J⁰ ⊆ F which almost satisfies the as- sumptions (set-up) of the Counting Lemma.

• “Massage” J⁰ to find a subgraph J ⊆ F satisfying the set-up of the Counting Lemma.

Outline of the Proof of the Main Result (continued)

• Consider the cluster graph F_Cluster of F with respect to P, depending on ε and α.

• Use (P2) and “smart” choice of constants to show that e(F_Cluster) ≥ 1 − 1

χ(H) − 1 + δ⁰

! t

2

.

⇒ H ⊆ F_Cluster by Erd ˝os–Stone–Simonovits

• This copy of H translates to a J⁰ ⊆ F which almost satisfies the as- sumptions (set-up) of the Counting Lemma.

• “Massage” J⁰ to find a subgraph J ⊆ F satisfying the set-up of the Counting Lemma.

⇒ H ⊆ F with probability 1 − o(1)

1 Introduction

Extremal Graph Theory Random Graph Theory

Extremal Random Graph Theory The Main Result

2 Proof of the Main Result

Szemer ´edi’s regularity lemma for sparse graphs

The Counting Lemma for subgraphs for random graphs Outline of the proof the Main Result

Summary & Outline

1 Introduction

Extremal Graph Theory Random Graph Theory

Extremal Random Graph Theory The Main Result

2 Proof of the Main Result

Szemer ´edi’s regularity lemma for sparse graphs

The Counting Lemma for subgraphs for random graphs Outline of the proof the Main Result

3 Proof of the Counting Lemma

Outline of the proof of the Counting Lemma for H = K₄ − e The k-tuple Lemma

The “Pick-Up” Lemma

4

−

Outline of the proof of the Counting Lemma for H = K

− e

4

−

Outline of the proof of the Counting Lemma for H = K

− e

4

−

Outline of the proof of the Counting Lemma for H = K

− e

4

−

Outline of the proof of the Counting Lemma for H = K

− e

4

−

Outline of the proof of the Counting Lemma for H = K

− e

4

−

Outline of the proof of the Counting Lemma for H = K

− e

4

−

Outline of the proof of the Counting Lemma for H = K

− e

4

−

Outline of the proof of the Counting Lemma for H = K

− e

The k-tuple Lemma

Let G ∈ G(n, p) and B = (U, W;E) be a bipartite, not necessarily induced subgraph of G with |U| = m1 and |W| = m2. Let q = e(B)/(m1m2).

The k-tuple Lemma

Let G ∈ G(n, p) and B = (U, W;E) be a bipartite, not necessarily induced subgraph of G with |U| = m1 and |W| = m2. Let q = e(B)/(m1m2). For some fixed integer k ≥ 1 and real γ > 0 consider

B^(k)(U, W;γ) = {b = {v1, . . . , v_k} ∈ [W]^k| |d_B(b)− q^km1| ≥ γq^km1} where d_B(b) is the size of the joint neighbourhood of b in B.

The k-tuple Lemma

Let G ∈ G(n, p) and B = (U, W;E) be a bipartite, not necessarily induced subgraph of G with |U| = m1 and |W| = m2. Let q = e(B)/(m1m2). For some fixed integer k ≥ 1 and real γ > 0 consider

B^(k)(U, W;γ) = {b = {v1, . . . , v_k} ∈ [W]^k| |d_B(b)− q^km1| ≥ γq^km1} where d_B(b) is the size of the joint neighbourhood of b in B.

Lemma (k-tuple Lemma; Kohayakawa and R ¨odl 2002+). For every reals α, γ, η > 0, integer k ≥ 1, and function m0 = m0(n) such that p^km0 (logn)⁴ there exists a real constant ε > 0 for which a.a.s. G in G(n, p) satisfies the following property:

The k-tuple Lemma

Let G ∈ G(n, p) and B = (U, W;E) be a bipartite, not necessarily induced subgraph of G with |U| = m1 and |W| = m2. Let q = e(B)/(m1m2). For some fixed integer k ≥ 1 and real γ > 0 consider

B^(k)(U, W;γ) = {b = {v1, . . . , v_k} ∈ [W]^k| |d_B(b)− q^km1| ≥ γq^km1} where d_B(b) is the size of the joint neighbourhood of b in B.

Lemma (k-tuple Lemma; Kohayakawa and R ¨odl 2002+). For every reals α, γ, η > 0, integer k ≥ 1, and function m0 = m0(n) such that p^km0 (logn)⁴ there exists a real constant ε > 0 for which a.a.s. G in G(n, p) satisfies the following property:

If B = (U, W;F) ⊆ G is such that (i) e(B) ≥ αe(G[U, W]),

(ii) B is (ε, p)-regular,

(iii) |U| = m1 ≥ m0 and |W| = m2 ≥ m0,

The k-tuple Lemma

Let G ∈ G(n, p) and B = (U, W;E) be a bipartite, not necessarily induced subgraph of G with |U| = m1 and |W| = m2. Let q = e(B)/(m1m2). For some fixed integer k ≥ 1 and real γ > 0 consider

B^(k)(U, W;γ) = {b = {v1, . . . , v_k} ∈ [W]^k| |d_B(b)− q^km1| ≥ γq^km1} where d_B(b) is the size of the joint neighbourhood of b in B.

Lemma (k-tuple Lemma; Kohayakawa and R ¨odl 2002+). For every reals α, γ, η > 0, integer k ≥ 1, and function m0 = m0(n) such that p^km0 (logn)⁴ there exists a real constant ε > 0 for which a.a.s. G in G(n, p) satisfies the following property:

If B = (U, W;F) ⊆ G is such that (i) e(B) ≥ αe(G[U, W]),

(ii) B is (ε, p)-regular,

(iii) |U| = m1 ≥ m0 and |W| = m2 ≥ m0, then

|B^(k)(U, W;γ)| ≤ η

_m2 k

.

The “Pick-Up” Lemma

Set-up.

• k ≥ 2 fixed integer and m sufficently large, 1/m q = q(m) < 1, and set T = qm

Set-up.

• k ≥ 2 fixed integer and m sufficently large, 1/m q = q(m) < 1, and set T = qm

• V₁, . . . , V_k pairwise disjoint sets of size m

The “Pick-Up” Lemma

Set-up.

• k ≥ 2 fixed integer and m sufficently large, 1/m q = q(m) < 1, and set T = qm

• V₁, . . . , V_k pairwise disjoint sets of size m

• Consider the uniform probability space Ω = V₁ × V_k

T

× · · · × V_k−11 × V_k T

Set-up.

• k ≥ 2 fixed integer and m sufficently large, 1/m q = q(m) < 1, and set T = qm

• V₁, . . . , V_k pairwise disjoint sets of size m

• Consider the uniform probability space Ω = V₁ × V_k

T

× · · · × V_k−11 × V_k T

• For R_i ∈ ^Vi×_T^Vk and v_k ∈ V_k set d_R

i(v_k) = |{v_i ∈ V_i| {v_i, v_k} ∈ R_i}|

The “Pick-Up” Lemma

Set-up.

• k ≥ 2 fixed integer and m sufficently large, 1/m q = q(m) < 1, and set T = qm

• V₁, . . . , V_k pairwise disjoint sets of size m

• Consider the uniform probability space Ω = V₁ × V_k

T

× · · · × V_k−11 × V_k T

• For R_i ∈ ^Vi×_T^Vk and v_k ∈ V_k set d_R

i(v_k) = |{v_i ∈ V_i| {v_i, v_k} ∈ R_i}|

• B ⊆ V₁ × · · · × V_k

Set-up.

• k ≥ 2 fixed integer and m sufficently large, 1/m q = q(m) < 1, and set T = qm

• V₁, . . . , V_k pairwise disjoint sets of size m

• Consider the uniform probability space Ω = V₁ × V_k

T

× · · · × V_k−11 × V_k T

• For R_i ∈ ^Vi×_T^Vk and v_k ∈ V_k set d_R

i(v_k) = |{v_i ∈ V_i| {v_i, v_k} ∈ R_i}|

• B ⊆ V₁ × · · · × V_k

Definition (Π(ζ, µ, K,B)). For ζ, µ, K > 0 and B ⊆ V₁ × · · · ×V_k we say Π(ζ, µ, K,B) holds for R = (R₁, . . . , R_k−1) ∈ Ω if

V˜_k(K) = {v_k ∈ V_k | d_R

i(v_k) ≤ Kqm,∀1 ≤ i < k} and

B(R) = {b = (v₁, . . . , v_k) ∈ B | v_k ∈ V˜_k ∧ {v_i, v_k} ∈ R_i,∀1 ≤ i < k} satisfy the inequalities

|V˜_k| ≥ (1 − µ)m

|B(R)| ≤ ζq^k−1m^k.

k

Π(ζ, µ, K,B) holds for R = (R₁, . . . , R_k−1) ∈ Ω if V˜_k(K) = {v_k ∈ V_k | d_R

i(v_k) ≤ Kqm,∀1 ≤ i < k} and

B(R) = {b = (v₁, . . . , v_k) ∈ B | v_k ∈ V˜_k ∧ {v_i, v_k} ∈ R_i,∀1 ≤ i < k} satisfy the inequalities

|V˜_k| ≥ (1 − µ)m

|B(R)| ≤ ζq^k−1m^k.

Lemma (Pick-Up Lemma). For every β, ζ, µ > 0 there exist η = η(β, ζ, µ) >

0, K = K(β, µ) > 0 and m₀ such that if m ≥ m₀ and

|B| ≤ ηm^k, then

P(Π(ζ, µ, K,B) fails for R ∈ Ω) ≤ β^(k−1)T.