Asymptotic Structure of Graphs with
the Minimum Number of Triangles
Oleg Pikhurko
University of Warwick
Erdős Centennial Conference
Erdős Lap Number
Erdős Lap Number
Erdős Lap Number
Lap Graph is Growing
Lap Graph is Growing
Lap Graph is Growing
Open Question
Open Question
Open Question (Małgorzata Bednarska, $10):
Is there a person with
Erdős Number = Erdős Lap Number = 1?
Open Question
Open Question (Małgorzata Bednarska, $10):
Is there a person with
Erdős Number = Erdős Lap Number = 1?
Open Question
Open Question (Małgorzata Bednarska, $10):
Is there a person with
Erdős Number = Erdős Lap Number = 1?
János Pach?
Open Question
Open Question (Małgorzata Bednarska, $10):
Is there a person with
Erdős Number = Erdős Lap Number = 1?
János Pach?
It’s not him on the photo!
Open Question
Open Question (Małgorzata Bednarska, $10):
Is there a person with
Erdős Number = Erdős Lap Number = 1?
János Pach?
It’s not him on the photo!
Still open...
Erdős-Rademacher Problem
Erdős-Rademacher Problem
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g(n, m) := min{#K3 (G) : v (G) = n, e(G) = m}
Erdős-Rademacher Problem
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g(n, m) := min{#K3 (G) : v (G) = n, e(G) = m}
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Mantel 1906, Turán’41: max{m : g(n, m) = 0} = b n4 c
2
Erdős-Rademacher Problem
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g(n, m) := min{#K3 (G) : v (G) = n, e(G) = m}
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Mantel 1906, Turán’41: max{m : g(n, m) = 0} = b n4 c
I
Rademacher’41: g(n, b n4 c + 1) = b n2 c
2
2
Just Above the Turán Function
Just Above the Turán Function
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2
Erdős’55: m ≤ b n4 c + 3
Just Above the Turán Function
2
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Erdős’55: m ≤ b n4 c + 3
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Erdős’62: m ≤ b n4 c + εn
2
Just Above the Turán Function
2
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Erdős’55: m ≤ b n4 c + 3
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Erdős’62: m ≤ b n4 c + εn
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Erdős’55: Is g(n, b n4 c + q) = q · b n2 c for q < n/2 ?
2
2
Just Above the Turán Function
2
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Erdős’55: m ≤ b n4 c + 3
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Erdős’62: m ≤ b n4 c + εn
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Erdős’55: Is g(n, b n4 c + q) = q · b n2 c for q < n/2 ?
2
2
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Kk ,k + q edges versus Kk +1,k −1 + (q + 1) edges
Just Above the Turán Function
2
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Erdős’55: m ≤ b n4 c + 3
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Erdős’62: m ≤ b n4 c + εn
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Erdős’55: Is g(n, b n4 c + q) = q · b n2 c for q < n/2 ?
2
2
I
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Kk ,k + q edges versus Kk +1,k −1 + (q + 1) edges
Lovász-Simonovits’75: Yes
Just Above the Turán Function
2
I
Erdős’55: m ≤ b n4 c + 3
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Erdős’62: m ≤ b n4 c + εn
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Erdős’55: Is g(n, b n4 c + q) = q · b n2 c for q < n/2 ?
2
2
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I
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Kk ,k + q edges versus Kk +1,k −1 + (q + 1) edges
Lovász-Simonovits’75: Yes
2
Lovász-Simonovits’83: m ≤ b n4 c + εn2
Asymptotic Version
Asymptotic Version
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g(a) := limn→∞
g(n,a(n2))
(n3)
Asymptotic Version
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g(a) := limn→∞
g(n,a(n2))
(n3)
Upper bound: complete partite graphs
Asymptotic Version
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g(a) := limn→∞
g(n,a(n2))
(n3)
Upper bound: complete partite graphs
Goodman bound: g(a) ≥ 2a2 − a
Asymptotic Version
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I
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g(a) := limn→∞
g(n,a(n2))
(n3)
Upper bound: complete partite graphs
Goodman bound: g(a) ≥ 2a2 − a
Moon-Moser’62, Nordhaus-Stewart’63, Bollobás’76...
Possible Edge/Triangle Densities (in Limit)
Possible Edge/Triangle Densities (in Limit)
K3
1.0
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
K2
Possible Edge/Triangle Densities (in Limit)
K3
K3
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.2
0.2
0.4
0.6
0.8
1.0
K2
0.4
0.6
0.8
1.0
K2
Determining g(a)
Determining g(a)
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Fisher’89:
1
2
≤a≤
2
3
Determining g(a)
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Fisher’89: 12 ≤ a ≤
Razborov’08: All a
2
3
Determining g(a)
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I
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Fisher’89: 12 ≤ a ≤ 23
Razborov’08: All a
Upper bound: Kcn,...,cn,(1−tc)n
Determining g(a)
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Fisher’89: 12 ≤ a ≤ 23
Razborov’08: All a
Upper bound: Kcn,...,cn,(1−tc)n
No stability
Determining g(a)
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Fisher’89: 12 ≤ a ≤ 23
Razborov’08: All a
Upper bound: Kcn,...,cn,(1−tc)n
No stability
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Hna : modify the last two parts
Determining g(a)
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Fisher’89: 12 ≤ a ≤ 23
Razborov’08: All a
Upper bound: Kcn,...,cn,(1−tc)n
No stability
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Hna : modify the last two parts
P.-Razborov ≥’13:
∀ almost extremal Gn is o(n2 )-close to some Hna
Graph Limits
Graph Limits
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Subgraph density
d(F , G) = Prob G[ random v (F )-set ] ∼
=F
Graph Limits
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Subgraph density
d(F , G) = Prob G[ random v (F )-set ] ∼
=F
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F = {finite graphs}
Graph Limits
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Subgraph density
d(F , G) = Prob G[ random v (F )-set ] ∼
=F
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F = {finite graphs}
(Gn ) converges if
∀ F ∈ F ∃ lim d(F , Gn )
n→∞
Graph Limits
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Subgraph density
d(F , G) = Prob G[ random v (F )-set ] ∼
=F
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F = {finite graphs}
(Gn ) converges if
∀ F ∈ F ∃ lim d(F , Gn ) =: φ(F )
n→∞
Graph Limits
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Subgraph density
d(F , G) = Prob G[ random v (F )-set ] ∼
=F
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F = {finite graphs}
(Gn ) converges if
∀ F ∈ F ∃ lim d(F , Gn ) =: φ(F )
n→∞
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LIM = {all such φ}
Graph Limits
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Subgraph density
d(F , G) = Prob G[ random v (F )-set ] ∼
=F
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F = {finite graphs}
(Gn ) converges if
∀ F ∈ F ∃ lim d(F , Gn ) =: φ(F )
n→∞
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LIM = {all such φ} ⊆ [0, 1]F
Extremal Limits
Extremal Limits
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Extremal limit: limits of almost extremal graphs
Extremal Limits
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Extremal limit: limits of almost extremal graphs
Equivalently: { φ ∈ LIM : φ(K3 ) = g(φ(K2 )) }
Extremal Limits
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Extremal limit: limits of almost extremal graphs
Equivalently: { φ ∈ LIM : φ(K3 ) = g(φ(K2 )) }
P.-Razborov ≥’13: {extremal limits} = {limits of Hna ’s}
Extremal Limits
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Extremal limit: limits of almost extremal graphs
Equivalently: { φ ∈ LIM : φ(K3 ) = g(φ(K2 )) }
P.-Razborov ≥’13: {extremal limits} = {limits of Hna ’s}
Implies the discrete theorem
Extremal Limits
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Extremal limit: limits of almost extremal graphs
Equivalently: { φ ∈ LIM : φ(K3 ) = g(φ(K2 )) }
P.-Razborov ≥’13: {extremal limits} = {limits of Hna ’s}
Implies the discrete theorem
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Cut distance
Extremal Limits
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Extremal limit: limits of almost extremal graphs
Equivalently: { φ ∈ LIM : φ(K3 ) = g(φ(K2 )) }
P.-Razborov ≥’13: {extremal limits} = {limits of Hna ’s}
Implies the discrete theorem
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Cut distance
Frieze-Kannan’90
Extremal Limits
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I
I
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Extremal limit: limits of almost extremal graphs
Equivalently: { φ ∈ LIM : φ(K3 ) = g(φ(K2 )) }
P.-Razborov ≥’13: {extremal limits} = {limits of Hna ’s}
Implies the discrete theorem
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I
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Cut distance
Frieze-Kannan’90
Lovász-Szegedy’06, Borgs et al’08...
Extremal Limits
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Extremal limit: limits of almost extremal graphs
Equivalently: { φ ∈ LIM : φ(K3 ) = g(φ(K2 )) }
P.-Razborov ≥’13: {extremal limits} = {limits of Hna ’s}
Implies the discrete theorem
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I
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Cut distance
Frieze-Kannan’90
Lovász-Szegedy’06, Borgs et al’08...
Close to Hna in cut-distance ⇒ close in edit distance
Razborov’s Proof for a ∈ [ 12 , 23 ]
Razborov’s Proof for a ∈ [ 12 , 23 ]
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h(a) = conjectured value
Razborov’s Proof for a ∈ [ 12 , 23 ]
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h(a) = conjectured value
LIM ⊆ [0, 1]F is closed
Razborov’s Proof for a ∈ [ 12 , 23 ]
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I
h(a) = conjectured value
LIM ⊆ [0, 1]F is closed ⇒ compact
Razborov’s Proof for a ∈ [ 12 , 23 ]
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I
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h(a) = conjectured value
LIM ⊆ [0, 1]F is closed ⇒ compact
f (φ) := φ(K3 ) − h(φ(K2 )) is continuous
Razborov’s Proof for a ∈ [ 12 , 23 ]
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I
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h(a) = conjectured value
LIM ⊆ [0, 1]F is closed ⇒ compact
f (φ) := φ(K3 ) − h(φ(K2 )) is continuous
∃ φ0 that minimises f on {φ ∈ LIM : 12 ≤ φ(K2 ) ≤ 23 }
Razborov’s Proof for a ∈ [ 12 , 23 ]
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I
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h(a) = conjectured value
LIM ⊆ [0, 1]F is closed ⇒ compact
f (φ) := φ(K3 ) − h(φ(K2 )) is continuous
∃ φ0 that minimises f on {φ ∈ LIM : 12 ≤ φ(K2 ) ≤ 23 }
a := φ0 (K2 )
Razborov’s Proof for a ∈ [ 12 , 23 ]
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I
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h(a) = conjectured value
LIM ⊆ [0, 1]F is closed ⇒ compact
f (φ) := φ(K3 ) − h(φ(K2 )) is continuous
∃ φ0 that minimises f on {φ ∈ LIM : 12 ≤ φ(K2 ) ≤ 23 }
a := φ0 (K2 )
c : e(Kcn,cn,(1−2c)n ) ≈ a n2
Razborov’s Proof for a ∈ [ 12 , 23 ]
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I
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h(a) = conjectured value
LIM ⊆ [0, 1]F is closed ⇒ compact
f (φ) := φ(K3 ) − h(φ(K2 )) is continuous
∃ φ0 that minimises f on {φ ∈ LIM : 12 ≤ φ(K2 ) ≤ 23 }
a := φ0 (K2 )
c : e(Kcn,cn,(1−2c)n ) ≈ a n2
Assume
1
2
<a<
2
3
(o/w done by Goodman)
Razborov’s Proof for a ∈ [ 12 , 23 ]
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h(a) = conjectured value
LIM ⊆ [0, 1]F is closed ⇒ compact
f (φ) := φ(K3 ) − h(φ(K2 )) is continuous
∃ φ0 that minimises f on {φ ∈ LIM : 12 ≤ φ(K2 ) ≤ 23 }
a := φ0 (K2 )
c : e(Kcn,cn,(1−2c)n ) ≈ a n2
Assume 12 < a < 23 (o/w done by Goodman)
h is differentiable at a
Razborov’s Proof for a ∈ [ 12 , 23 ]
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I
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h(a) = conjectured value
LIM ⊆ [0, 1]F is closed ⇒ compact
f (φ) := φ(K3 ) − h(φ(K2 )) is continuous
∃ φ0 that minimises f on {φ ∈ LIM : 12 ≤ φ(K2 ) ≤ 23 }
a := φ0 (K2 )
c : e(Kcn,cn,(1−2c)n ) ≈ a n2
Assume 12 < a < 23 (o/w done by Goodman)
h is differentiable at a
Pick Gn → φ0
Razborov’s Proof for a ∈ [ 12 , 23 ]
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h(a) = conjectured value
LIM ⊆ [0, 1]F is closed ⇒ compact
f (φ) := φ(K3 ) − h(φ(K2 )) is continuous
∃ φ0 that minimises f on {φ ∈ LIM : 12 ≤ φ(K2 ) ≤ 23 }
a := φ0 (K2 )
c : e(Kcn,cn,(1−2c)n ) ≈ a n2
Assume 12 < a < 23 (o/w done by Goodman)
h is differentiable at a
Pick Gn → φ0
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Rate of growth: ≈ cn triangles per new edge
Razborov’s Proof for a ∈ [ 12 , 23 ]
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h(a) = conjectured value
LIM ⊆ [0, 1]F is closed ⇒ compact
f (φ) := φ(K3 ) − h(φ(K2 )) is continuous
∃ φ0 that minimises f on {φ ∈ LIM : 12 ≤ φ(K2 ) ≤ 23 }
a := φ0 (K2 )
c : e(Kcn,cn,(1−2c)n ) ≈ a n2
Assume 12 < a < 23 (o/w done by Goodman)
h is differentiable at a
Pick Gn → φ0
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Rate of growth: ≈ cn triangles per new edge
Gn has / cn triangles on almost every edge
At Most cn Triangles per Edge
At Most cn Triangles per Edge
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Flag algebra statement
φE0 (K3E ) ≤ c
a.s.
At Most cn Triangles per Edge
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Flag algebra statement
φE0 (K3E ) ≤ c
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Informal explanation:
a.s.
At Most cn Triangles per Edge
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Flag algebra statement
φE0 (K3E ) ≤ c
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Informal explanation:
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G n → φ0
a.s.
At Most cn Triangles per Edge
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Flag algebra statement
φE0 (K3E ) ≤ c
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a.s.
Informal explanation:
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G n → φ0
φE0 : Two random adjacent roots x1 , x2 in Gn
At Most cn Triangles per Edge
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Flag algebra statement
φE0 (K3E ) ≤ c
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a.s.
Informal explanation:
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G n → φ0
φE0 : Two random adjacent roots x1 , x2 in Gn
K3E : Density of rooted triangles
Vertex Removal
Vertex Removal
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Remove x ∈ V (Gn ):
Vertex Removal
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Remove x ∈ V (Gn ):
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∂ d(K2 , Gn ) :
Vertex Removal
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Remove x ∈ V (Gn ):
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∂ d(K2 , Gn ) :
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Remove edges: −d(x) n2
Vertex Removal
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Remove x ∈ V (Gn ):
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∂ d(K2 , Gn ) :
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I
Remove edges: −d(x) n2
n−1
Remove isolated x: × n2
=1+
2
2
n
+ ...
Vertex Removal
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Remove x ∈ V (Gn ):
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∂ d(K2 , Gn ) :
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I
I
Remove edges: −d(x) n2
n−1
Remove isolated x: × n2
=1+
2
Total change: −K21 (x)/ n2 + a n2 + ...
2
n
+ ...
Vertex Removal
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Remove x ∈ V (Gn ):
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∂ d(K2 , Gn ) :
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I
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I
Remove edges: −d(x) n2
n−1
Remove isolated x: × n2
=1+
2
Total change: −K21 (x)/ n2 + a n2 + ...
∂ d(K3 , Gn ) = −K31 (x)/
n
3
2
n
+ ...
+ φ0 (K3 ) n3 + ...
Vertex Removal
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Remove x ∈ V (Gn ):
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∂ d(K2 , Gn ) :
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I
I
I
I
Remove edges: −d(x) n2
n−1
Remove isolated x: × n2
=1+
2
Total change: −K21 (x)/ n2 + a n2 + ...
∂ d(K3 , Gn ) = −K31 (x)/
n
3
2
n
+ ...
+ φ0 (K3 ) n3 + ...
Expect: ∂d(K3 ) ' h0 (a) ∂d(K2 )
Vertex Removal
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Remove x ∈ V (Gn ):
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∂ d(K2 , Gn ) :
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I
I
I
I
I
Remove edges: −d(x) n2
n−1
Remove isolated x: × n2
=1+
2
Total change: −K21 (x)/ n2 + a n2 + ...
∂ d(K3 , Gn ) = −K31 (x)/
n
3
2
n
+ ...
+ φ0 (K3 ) n3 + ...
Expect: ∂d(K3 ) ' h0 (a) ∂d(K2 )
Cloning x: signs change
Vertex Removal
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Remove x ∈ V (Gn ):
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∂ d(K2 , Gn ) :
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I
I
I
I
I
I
Remove edges: −d(x) n2
n−1
Remove isolated x: × n2
=1+
2
Total change: −K21 (x)/ n2 + a n2 + ...
∂ d(K3 , Gn ) = −K31 (x)/
n
3
2
n
+ ...
+ φ0 (K3 ) n3 + ...
Expect: ∂d(K3 ) ' h0 (a) ∂d(K2 )
Cloning x: signs change
Approximate equality for almost all x
Vertex Removal
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Remove x ∈ V (Gn ):
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∂ d(K2 , Gn ) :
I
I
I
I
I
I
I
I
Remove edges: −d(x) n2
n−1
Remove isolated x: × n2
=1+
2
Total change: −K21 (x)/ n2 + a n2 + ...
∂ d(K3 , Gn ) = −K31 (x)/
n
3
2
n
+ ...
+ φ0 (K3 ) n3 + ...
Expect: ∂d(K3 ) ' h0 (a) ∂d(K2 )
Cloning x: signs change
Approximate equality for almost all x
Flag algebra statement:
−3! φ10 (K31 ) + 3φ0 (K3 ) = 3c −2φ10 (K21 ) + 2a
a.s.
Finishing line
Finishing line
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Recall: A.s.
Finishing line
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Recall: A.s.
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−3! φ10 (K31 ) + 3φ0 (K3 ) = 3c −2φ10 (K21 ) + 2a
Finishing line
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Recall: A.s.
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−3! φ10 (K31 ) + 3φ0 (K3 ) = 3c −2φ10 (K21 ) + 2a
φE0 (K3E ) ≤ c
Finishing line
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Recall: A.s.
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−3! φ10 (K31 ) + 3φ0 (K3 ) = 3c −2φ10 (K21 ) + 2a
φE0 (K3E ) ≤ c
Average?
Finishing line
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Recall: A.s.
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−3! φ10 (K31 ) + 3φ0 (K3 ) = 3c −2φ10 (K21 ) + 2a
φE0 (K3E ) ≤ c
Average?
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0=0
Finishing line
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Recall: A.s.
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−3! φ10 (K31 ) + 3φ0 (K3 ) = 3c −2φ10 (K21 ) + 2a
φE0 (K3E ) ≤ c
Average?
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0=0
/
Finishing line
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Recall: A.s.
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−3! φ10 (K31 ) + 3φ0 (K3 ) = 3c −2φ10 (K21 ) + 2a
φE0 (K3E ) ≤ c
Average?
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0=0
Slack
/
Finishing line
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Recall: A.s.
I
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−3! φ10 (K31 ) + 3φ0 (K3 ) = 3c −2φ10 (K21 ) + 2a
φE0 (K3E ) ≤ c
Average?
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0=0
Slack
/
/
Finishing line
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Recall: A.s.
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Average?
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−3! φ10 (K31 ) + 3φ0 (K3 ) = 3c −2φ10 (K21 ) + 2a
φE0 (K3E ) ≤ c
0=0
Slack
/
/
E
Multiply by K21 & P 3 and then average!
Finishing line
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Recall: A.s.
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Average?
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−3! φ10 (K31 ) + 3φ0 (K3 ) = 3c −2φ10 (K21 ) + 2a
φE0 (K3E ) ≤ c
0=0
Slack
/
/
E
Multiply by K21 & P 3 and then average!
Calculations give
Finishing line
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Recall: A.s.
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Average?
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−3! φ10 (K31 ) + 3φ0 (K3 ) = 3c −2φ10 (K21 ) + 2a
φE0 (K3E ) ≤ c
0=0
Slack
/
/
E
Multiply by K21 & P 3 and then average!
Calculations give
φ0 (K3 ) ≥
3ac(2a − 1) + φ0 (K4 ) + 14 φ0 (K 1,3 )
3c + 3a − 2
Finishing line
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Recall: A.s.
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0=0
Slack
/
/
E
Multiply by K21 & P 3 and then average!
Calculations give
φ0 (K3 ) ≥
I
Average?
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−3! φ10 (K31 ) + 3φ0 (K3 ) = 3c −2φ10 (K21 ) + 2a
φE0 (K3E ) ≤ c
3ac(2a − 1) + φ0 (K4 ) + 14 φ0 (K 1,3 )
3c + 3a − 2
φ0 (K4 ) ≥ 0 & φ0 (K 1,3 ) ≥ 0 ⇒ φ0 (K3 ) ≥ h(a)
Finishing line
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Recall: A.s.
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0=0
Slack
/
/
E
Multiply by K21 & P 3 and then average!
Calculations give
φ0 (K3 ) ≥
I
Average?
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I
−3! φ10 (K31 ) + 3φ0 (K3 ) = 3c −2φ10 (K21 ) + 2a
φE0 (K3E ) ≤ c
3ac(2a − 1) + φ0 (K4 ) + 14 φ0 (K 1,3 )
3c + 3a − 2
φ0 (K4 ) ≥ 0 & φ0 (K 1,3 ) ≥ 0 ⇒ φ0 (K3 ) ≥ h(a)
,
Structure of Extremal φ0
Structure of Extremal φ0
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Assume φ0 (K3 ) = h(a)
Structure of Extremal φ0
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Assume φ0 (K3 ) = h(a)
Lovász-Simonovits’83: a ∈ ( 21 , 23 )
Structure of Extremal φ0
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Assume φ0 (K3 ) = h(a)
Lovász-Simonovits’83: a ∈ ( 21 , 23 )
Density of K4 and K 1,3 is 0
Structure of Extremal φ0
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Assume φ0 (K3 ) = h(a)
Lovász-Simonovits’83: a ∈ ( 21 , 23 )
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Density of K4 and K 1,3 is 0
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If φ0 (P 3 ) = 0,
Structure of Extremal φ0
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Assume φ0 (K3 ) = h(a)
Lovász-Simonovits’83: a ∈ ( 21 , 23 )
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Density of K4 and K 1,3 is 0
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If φ0 (P 3 ) = 0,
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Complete partite
Structure of Extremal φ0
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Assume φ0 (K3 ) = h(a)
Lovász-Simonovits’83: a ∈ ( 21 , 23 )
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Density of K4 and K 1,3 is 0
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If φ0 (P 3 ) = 0,
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Complete partite
K4 -free
Structure of Extremal φ0
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Assume φ0 (K3 ) = h(a)
Lovász-Simonovits’83: a ∈ ( 21 , 23 )
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Density of K4 and K 1,3 is 0
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If φ0 (P 3 ) = 0,
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Complete partite
K4 -free ⇒ at most 3 parts
Structure of Extremal φ0
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I
Assume φ0 (K3 ) = h(a)
Lovász-Simonovits’83: a ∈ ( 21 , 23 )
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Density of K4 and K 1,3 is 0
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If φ0 (P 3 ) = 0,
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Complete partite
K4 -free ⇒ at most 3 parts ⇒ done!
Case 2: φ0(P 3) > 0
Case 2: φ0(P 3) > 0
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Special graphs F1 and F2 :
Case 2: φ0(P 3) > 0
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Special graphs F1 and F2 :
Claim: φ0 (F1 ) = φ0 (F2 ) = 0
Case 2: φ0(P 3) > 0
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Special graphs F1 and F2 :
Claim: φ0 (F1 ) = φ0 (F2 ) = 0
Claim: Exist many P 3 ’s st
Case 2: φ0(P 3) > 0
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Special graphs F1 and F2 :
Claim: φ0 (F1 ) = φ0 (F2 ) = 0
Claim: Exist many P 3 ’s st
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|A| = Ω(n): vertices sending 3 edges to it
Case 2: φ0(P 3) > 0
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Special graphs F1 and F2 :
Claim: φ0 (F1 ) = φ0 (F2 ) = 0
Claim: Exist many P 3 ’s st
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|A| = Ω(n): vertices sending 3 edges to it
|B| = Ω(n): vertices sending ≤ 2 edges to it
Case 2: φ0(P 3) > 0
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Special graphs F1 and F2 :
Claim: φ0 (F1 ) = φ0 (F2 ) = 0
Claim: Exist many P 3 ’s st
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|A| = Ω(n): vertices sending 3 edges to it
|B| = Ω(n): vertices sending ≤ 2 edges to it
Non-edge across → a copy of F1 , F2 , or K 1,3
Case 2: φ0(P 3) > 0
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Special graphs F1 and F2 :
Claim: φ0 (F1 ) = φ0 (F2 ) = 0
Claim: Exist many P 3 ’s st
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|A| = Ω(n): vertices sending 3 edges to it
|B| = Ω(n): vertices sending ≤ 2 edges to it
Non-edge across → a copy of F1 , F2 , or K 1,3
Gn [A, B] is almost complete
Case 2: φ0(P 3) > 0
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Special graphs F1 and F2 :
Claim: φ0 (F1 ) = φ0 (F2 ) = 0
Claim: Exist many P 3 ’s st
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I
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|A| = Ω(n): vertices sending 3 edges to it
|B| = Ω(n): vertices sending ≤ 2 edges to it
Non-edge across → a copy of F1 , F2 , or K 1,3
Gn [A, B] is almost complete
Induction + calculations
,
Clique Minimisation Problem
Clique Minimisation Problem
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Open: Exact result for K3
Clique Minimisation Problem
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Open: Exact result for K3
Nikiforov’11: Asymptotic solution for K4
Clique Minimisation Problem
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Open: Exact result for K3
Nikiforov’11: Asymptotic solution for K4
Reiher ≥’13: Asymptotic solution for Kr
Clique Minimisation Problem
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Open: Exact result for K3
Nikiforov’11: Asymptotic solution for K4
Reiher ≥’13: Asymptotic solution for Kr
Open: Structure & exact result
General Graphs
General Graphs
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Colour critical: χ(F ) = r + 1 & χ(F − e) = r
General Graphs
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Colour critical: χ(F ) = r + 1 & χ(F − e) = r
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Simonovits’68: ex(n, F ) = ex(n, Kr +1 ), n ≥ n0
General Graphs
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Colour critical: χ(F ) = r + 1 & χ(F − e) = r
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Simonovits’68: ex(n, F ) = ex(n, Kr +1 ), n ≥ n0
Mubayi’10: Asymptotic for m ≤ ex(n, F ) + εF n
General Graphs
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Colour critical: χ(F ) = r + 1 & χ(F − e) = r
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Simonovits’68: ex(n, F ) = ex(n, Kr +1 ), n ≥ n0
Mubayi’10: Asymptotic for m ≤ ex(n, F ) + εF n
P.-Yilma ≥’13: Asymptotic for m ≤ ex(n, F ) + o(n2 )
General Graphs
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Colour critical: χ(F ) = r + 1 & χ(F − e) = r
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Simonovits’68: ex(n, F ) = ex(n, Kr +1 ), n ≥ n0
Mubayi’10: Asymptotic for m ≤ ex(n, F ) + εF n
P.-Yilma ≥’13: Asymptotic for m ≤ ex(n, F ) + o(n2 )
Bipartite F
General Graphs
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Colour critical: χ(F ) = r + 1 & χ(F − e) = r
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Simonovits’68: ex(n, F ) = ex(n, Kr +1 ), n ≥ n0
Mubayi’10: Asymptotic for m ≤ ex(n, F ) + εF n
P.-Yilma ≥’13: Asymptotic for m ≤ ex(n, F ) + o(n2 )
Bipartite F
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Conjecture (Erdős-Simonovits’82, Sidorenko’93):
General Graphs
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Colour critical: χ(F ) = r + 1 & χ(F − e) = r
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Simonovits’68: ex(n, F ) = ex(n, Kr +1 ), n ≥ n0
Mubayi’10: Asymptotic for m ≤ ex(n, F ) + εF n
P.-Yilma ≥’13: Asymptotic for m ≤ ex(n, F ) + o(n2 )
Bipartite F
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Conjecture (Erdős-Simonovits’82, Sidorenko’93):
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Random graphs
General Graphs
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Colour critical: χ(F ) = r + 1 & χ(F − e) = r
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Simonovits’68: ex(n, F ) = ex(n, Kr +1 ), n ≥ n0
Mubayi’10: Asymptotic for m ≤ ex(n, F ) + εF n
P.-Yilma ≥’13: Asymptotic for m ≤ ex(n, F ) + o(n2 )
Bipartite F
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Conjecture (Erdős-Simonovits’82, Sidorenko’93):
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Random graphs
..., Conlon-Fox-Sudakov’10, Li-Szegedy ≥’13, ...
Thank you!
Photos: Math PUrview, Gil Kalai’s blog & Erdős Lap Number ,