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Critical points in zero-one laws for G(n, p)
Maksim Zhukovskii
MSU, MIPT, Yandex
Workshop on random graphs and their applications
26 October 2013
1/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 2: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/2.jpg)
First-order properties.
First-order formulae:relational symbols ∼,=;logical connectivities ¬,⇒,⇔,∨,∧;variables x, y, x1, ...;quanti�ers ∀,∃.
• property of a graph to be complete
∀x∀y (¬(x = y)⇒ (x ∼ y)).
• property of a graph to contain a triangle
∃x ∃y ∃z ((x ∼ y) ∧ (y ∼ z) ∧ (x ∼ z)).
• property of a graph to have chromatic number k
2/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 3: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/3.jpg)
First-order properties.
First-order formulae:relational symbols ∼,=;logical connectivities ¬,⇒,⇔,∨,∧;variables x, y, x1, ...;quanti�ers ∀,∃.
• property of a graph to be complete
∀x∀y (¬(x = y)⇒ (x ∼ y)).
• property of a graph to contain a triangle
∃x ∃y ∃z ((x ∼ y) ∧ (y ∼ z) ∧ (x ∼ z)).
• property of a graph to have chromatic number k
2/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 4: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/4.jpg)
First-order properties.
First-order formulae:relational symbols ∼,=;logical connectivities ¬,⇒,⇔,∨,∧;variables x, y, x1, ...;quanti�ers ∀,∃.
• property of a graph to be complete
∀x∀y (¬(x = y)⇒ (x ∼ y)).
• property of a graph to contain a triangle
∃x ∃y ∃z ((x ∼ y) ∧ (y ∼ z) ∧ (x ∼ z)).
• property of a graph to have chromatic number k
2/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 5: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/5.jpg)
First-order properties.
First-order formulae:relational symbols ∼,=;logical connectivities ¬,⇒,⇔,∨,∧;variables x, y, x1, ...;quanti�ers ∀,∃.
• property of a graph to be complete
∀x∀y (¬(x = y)⇒ (x ∼ y)).
• property of a graph to contain a triangle
∃x ∃y ∃z ((x ∼ y) ∧ (y ∼ z) ∧ (x ∼ z)).
• property of a graph to have chromatic number k
2/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 6: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/6.jpg)
Limit probabilities.
G(n, p): n � number of vertices, p � edge appearanceprobability
G � an arbitrary graph, LG � property of containing G.LG is the �rst order property.
maximal density ρmax(G) = maxH⊆G{ρ(H)}, ρ(H) = e(H)v(H)
,
p0 = n−1/ρmax(G);
p�p0 ⇒ limn→∞
Pn,p(LG) = 1,
p�p0 ⇒ limn→∞
Pn,p(LG) = 0,
p=p0 ⇒ limn→∞
Pn,p(LG)/∈ {0, 1}.
If G � set of graphs, L = {LG, LG, G ∈ G},p = n−α, α is irrational
then ∀L ∈ L limn→∞ Pn,p(L) ∈ {0, 1}.
3/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 7: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/7.jpg)
Limit probabilities.
G(n, p): n � number of vertices, p � edge appearanceprobability
G � an arbitrary graph, LG � property of containing G.
LG is the �rst order property.
maximal density ρmax(G) = maxH⊆G{ρ(H)}, ρ(H) = e(H)v(H)
,
p0 = n−1/ρmax(G);
p�p0 ⇒ limn→∞
Pn,p(LG) = 1,
p�p0 ⇒ limn→∞
Pn,p(LG) = 0,
p=p0 ⇒ limn→∞
Pn,p(LG)/∈ {0, 1}.
If G � set of graphs, L = {LG, LG, G ∈ G},p = n−α, α is irrational
then ∀L ∈ L limn→∞ Pn,p(L) ∈ {0, 1}.
3/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 8: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/8.jpg)
Limit probabilities.
G(n, p): n � number of vertices, p � edge appearanceprobability
G � an arbitrary graph, LG � property of containing G.LG is the �rst order property.
maximal density ρmax(G) = maxH⊆G{ρ(H)}, ρ(H) = e(H)v(H)
,
p0 = n−1/ρmax(G);
p�p0 ⇒ limn→∞
Pn,p(LG) = 1,
p�p0 ⇒ limn→∞
Pn,p(LG) = 0,
p=p0 ⇒ limn→∞
Pn,p(LG)/∈ {0, 1}.
If G � set of graphs, L = {LG, LG, G ∈ G},p = n−α, α is irrational
then ∀L ∈ L limn→∞ Pn,p(L) ∈ {0, 1}.
3/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 9: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/9.jpg)
Limit probabilities.
G(n, p): n � number of vertices, p � edge appearanceprobability
G � an arbitrary graph, LG � property of containing G.LG is the �rst order property.
maximal density ρmax(G) = maxH⊆G{ρ(H)}, ρ(H) = e(H)v(H)
,
p0 = n−1/ρmax(G);
p�p0 ⇒ limn→∞
Pn,p(LG) = 1,
p�p0 ⇒ limn→∞
Pn,p(LG) = 0,
p=p0 ⇒ limn→∞
Pn,p(LG)/∈ {0, 1}.
If G � set of graphs, L = {LG, LG, G ∈ G},p = n−α, α is irrational
then ∀L ∈ L limn→∞ Pn,p(L) ∈ {0, 1}.
3/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 10: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/10.jpg)
Limit probabilities.
G(n, p): n � number of vertices, p � edge appearanceprobability
G � an arbitrary graph, LG � property of containing G.LG is the �rst order property.
maximal density ρmax(G) = maxH⊆G{ρ(H)}, ρ(H) = e(H)v(H)
,
p0 = n−1/ρmax(G);
p�p0 ⇒ limn→∞
Pn,p(LG) = 1,
p�p0 ⇒ limn→∞
Pn,p(LG) = 0,
p=p0 ⇒ limn→∞
Pn,p(LG)/∈ {0, 1}.
If G � set of graphs, L = {LG, LG, G ∈ G},p = n−α, α is irrational
then ∀L ∈ L limn→∞ Pn,p(L) ∈ {0, 1}.
3/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 11: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/11.jpg)
Limit probabilities.
G(n, p): n � number of vertices, p � edge appearanceprobability
G � an arbitrary graph, LG � property of containing G.LG is the �rst order property.
maximal density ρmax(G) = maxH⊆G{ρ(H)}, ρ(H) = e(H)v(H)
,
p0 = n−1/ρmax(G);
p�p0 ⇒ limn→∞
Pn,p(LG) = 1,
p�p0 ⇒ limn→∞
Pn,p(LG) = 0,
p=p0 ⇒ limn→∞
Pn,p(LG)/∈ {0, 1}.
If G � set of graphs, L = {LG, LG, G ∈ G},p = n−α, α is irrational
then ∀L ∈ L limn→∞ Pn,p(L) ∈ {0, 1}.
3/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 12: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/12.jpg)
Limit probabilities.
G(n, p): n � number of vertices, p � edge appearanceprobability
G � an arbitrary graph, LG � property of containing G.LG is the �rst order property.
maximal density ρmax(G) = maxH⊆G{ρ(H)}, ρ(H) = e(H)v(H)
,
p0 = n−1/ρmax(G);
p�p0 ⇒ limn→∞
Pn,p(LG) = 1,
p�p0 ⇒ limn→∞
Pn,p(LG) = 0,
p=p0 ⇒ limn→∞
Pn,p(LG)/∈ {0, 1}.
If G � set of graphs, L = {LG, LG, G ∈ G},p = n−α, α is irrational
then ∀L ∈ L limn→∞ Pn,p(L) ∈ {0, 1}.
3/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 13: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/13.jpg)
Zero-one law
L � the set of all �rst-order properties.
De�nition
The random graph obeys zero-one law if ∀L ∈ L
limn→∞
Pn,p(n)(L) ∈ {0, 1}.
P is a set of functions p such that G(n, p) obeys zero-one law.
Theorem [J.H. Spencer, S. Shelah, 1988]
Let p = n−α, α ∈ (0, 1].
If α ∈ R \Q then p ∈ P.If α ∈ Q then p /∈ P.
4/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Zero-one law
L � the set of all �rst-order properties.
De�nition
The random graph obeys zero-one law if ∀L ∈ L
limn→∞
Pn,p(n)(L) ∈ {0, 1}.
P is a set of functions p such that G(n, p) obeys zero-one law.
Theorem [J.H. Spencer, S. Shelah, 1988]
Let p = n−α, α ∈ (0, 1].
If α ∈ R \Q then p ∈ P.If α ∈ Q then p /∈ P.
4/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Quanti�er depth.
Quanti�er depths of x = y, x ∼ y equal 0.
If quanti�er depth of φ equals k then quanti�er depth of ¬φequals k as well.
If quanti�er depth of φ1 equals k1, quanti�er depth of φ2 equalsk2 then quanti�er depths of φ1 ⇒ φ2, φ1 ⇔ φ2, φ1 ∨ φ2,φ1 ∧ φ2 equal max{k1, k2}.
If quanti�er depth of φ(x) equals k, then quanti�er depths of∃xφ(x), ∀xφ(x) equal k + 1.
5/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Quanti�er depth.
Quanti�er depths of x = y, x ∼ y equal 0.
If quanti�er depth of φ equals k then quanti�er depth of ¬φequals k as well.
If quanti�er depth of φ1 equals k1, quanti�er depth of φ2 equalsk2 then quanti�er depths of φ1 ⇒ φ2, φ1 ⇔ φ2, φ1 ∨ φ2,φ1 ∧ φ2 equal max{k1, k2}.
If quanti�er depth of φ(x) equals k, then quanti�er depths of∃xφ(x), ∀xφ(x) equal k + 1.
5/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Quanti�er depth.
Quanti�er depths of x = y, x ∼ y equal 0.
If quanti�er depth of φ equals k then quanti�er depth of ¬φequals k as well.
If quanti�er depth of φ1 equals k1, quanti�er depth of φ2 equalsk2 then quanti�er depths of φ1 ⇒ φ2, φ1 ⇔ φ2, φ1 ∨ φ2,φ1 ∧ φ2 equal max{k1, k2}.
If quanti�er depth of φ(x) equals k, then quanti�er depths of∃xφ(x), ∀xφ(x) equal k + 1.
5/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 18: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/18.jpg)
Quanti�er depth.
Quanti�er depths of x = y, x ∼ y equal 0.
If quanti�er depth of φ equals k then quanti�er depth of ¬φequals k as well.
If quanti�er depth of φ1 equals k1, quanti�er depth of φ2 equalsk2 then quanti�er depths of φ1 ⇒ φ2, φ1 ⇔ φ2, φ1 ∨ φ2,φ1 ∧ φ2 equal max{k1, k2}.
If quanti�er depth of φ(x) equals k, then quanti�er depths of∃xφ(x), ∀xφ(x) equal k + 1.
5/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Bounded quanti�er depth.
Lk � the class of �rst-order properties de�ned by formulae withquanti�er depth bounded by k.
Random graph G(n, p) obeys zero-one k-law if ∀L ∈ LklimN→∞
Pn,p(n)(L) ∈ {0, 1}.
Pk � the class of all functions p = p(n) such that random graphG(n, p) obeys zero-one k-law.
Example: (∀x ∃y (x ∼ y)) ∧ (∀x ∃y ¬(x ∼ y)) k = 2∀x∃y ∃z ((x ∼ y) ∧ (¬(x ∼ z))) k = 3
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Bounded quanti�er depth.
Lk � the class of �rst-order properties de�ned by formulae withquanti�er depth bounded by k.
Random graph G(n, p) obeys zero-one k-law if ∀L ∈ LklimN→∞
Pn,p(n)(L) ∈ {0, 1}.
Pk � the class of all functions p = p(n) such that random graphG(n, p) obeys zero-one k-law.
Example: (∀x ∃y (x ∼ y)) ∧ (∀x ∃y ¬(x ∼ y)) k = 2∀x ∃y ∃z ((x ∼ y) ∧ (¬(x ∼ z))) k = 3
6/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Zero-one k-laws
Zhukovskii; 2010 (zero-one k-law)
Let p = n−α, k ∈ N, k ≥ 3.
If 0 < α < 1k−2 then p ∈ Pk.
Zhukovskii; 2012+ (extension of zero-one k-law)
Let p = n−α, k ∈ N, k ≥ 4, Q = {ab , a, b ∈ N, a ≤ 2k−1}
If α = 1− 12k−1+β
, β ∈ (0,∞) \ Q then p ∈ Pk.
(1−
1
2k, 1
)⋃(1−
1
2k − 1, 1−
1
2k
)⋃. . .⋃
(1−
1
2k−1 + 2k−2, 1−
1
2k−1 + 2k−2 + 1
)⋃(1−
1
2k−1 + 2k−1−12
, 1−1
2k−1 + 2k−2
)
⋃. . .⋃1−
1
2k−1 +2k−1−
[2k−1
3
]2
, 1−1
2k−1 + 2k−1
3
⋃ . . .? . . .⋃[
0,1
k − 2
).
7/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Zero-one k-laws
Zhukovskii; 2010 (zero-one k-law)
Let p = n−α, k ∈ N, k ≥ 3.
If 0 < α < 1k−2 then p ∈ Pk.
Zhukovskii; 2012+ (extension of zero-one k-law)
Let p = n−α, k ∈ N, k ≥ 4, Q = {ab , a, b ∈ N, a ≤ 2k−1}
If α = 1− 12k−1+β
, β ∈ (0,∞) \ Q then p ∈ Pk.
(1−
1
2k, 1
)⋃(1−
1
2k − 1, 1−
1
2k
)⋃. . .⋃
(1−
1
2k−1 + 2k−2, 1−
1
2k−1 + 2k−2 + 1
)⋃(1−
1
2k−1 + 2k−1−12
, 1−1
2k−1 + 2k−2
)
⋃. . .⋃1−
1
2k−1 +2k−1−
[2k−1
3
]2
, 1−1
2k−1 + 2k−1
3
⋃ . . .? . . .⋃[
0,1
k − 2
).
7/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 23: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/23.jpg)
Zero-one k-laws
Zhukovskii; 2010 (zero-one k-law)
Let p = n−α, k ∈ N, k ≥ 3.
If 0 < α < 1k−2 then p ∈ Pk.
Zhukovskii; 2012+ (extension of zero-one k-law)
Let p = n−α, k ∈ N, k ≥ 4, Q = {ab , a, b ∈ N, a ≤ 2k−1}
If α = 1− 12k−1+β
, β ∈ (0,∞) \ Q then p ∈ Pk.
(1−
1
2k, 1
)⋃(1−
1
2k − 1, 1−
1
2k
)⋃. . .⋃
(1−
1
2k−1 + 2k−2, 1−
1
2k−1 + 2k−2 + 1
)⋃(1−
1
2k−1 + 2k−1−12
, 1−1
2k−1 + 2k−2
)
⋃. . .⋃1−
1
2k−1 +2k−1−
[2k−1
3
]2
, 1−1
2k−1 + 2k−1
3
⋃ . . .? . . .⋃[
0,1
k − 2
).
7/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Zero-one k-laws
Zhukovskii; 2013++ (no k-law)
Let p = n−α, k ∈ N.
If k ≥ 3, α = 1k−2 then p /∈ Pk.
If k ≥ 4,Q̃ =
2k−1 − 2 · 1, . . . , 1;2k−1−2·1−1·2
2, . . . , 1
2; 2k−1−2·1−2·2
3, . . . , 1
3; 2k−1−2·1−3·2
4, . . . , 1
4;
2k−1−2·1−3·2−1·35
, . . . ; . . . ; 2k−1−2·1−3·2−4·38
, . . . ;. . . ; . . . ; . . .
α = 1− 1
2k−1+β, β ∈ Q̃ then p /∈ Pk.
8/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Critical points
1− α : 0→ 1
9/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Critical points
1− α : 0→ 1
0
9/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Critical points
1− α : 0→ 1
0 12k
9/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Critical points
1− α : 0→ 1
0 12k
12k−1
9/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Critical points
1− α : 0→ 1
0 12k
12k−1
12k−2
9/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Critical points
1− α : 0→ 1
0 12k
12k−1
12k−2
12k−3
9/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Critical points
1− α : 0→ 1
0 12k
12k−1
12k−2
12k−3
. . . 12k−2k−2
9/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Critical points
1− α : 0→ 1
0 12k
12k−1
12k−2
12k−3
. . . 12k−2k−2 . . .
9/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Critical points
1− α : 0→ 1
0 12k
12k−1
12k−2
12k−3
. . . 12k−2k−2 . . . ?
Large gap: [1/2k−1, 1− 1/(k − 2))
9/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 34: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/34.jpg)
Critical points
1− α : 0→ 1
0 12k
12k−1
12k−2
12k−3
. . . 12k−2k−2 . . . ? 1− 1
k−2
Large gap: [1/2k−1, 1− 1/(k − 2))
9/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 35: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/35.jpg)
Critical points
1− α : 0→ 1
0 12k−2
12k−3
. . . 12k−2k−2 . . . ? 1− 1
k−2
Large gap: [1/2k−1, 1− 1/(k − 2))
9/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 36: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/36.jpg)
Intervals
10/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 37: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/37.jpg)
Ehrenfeucht game
EHR(G,H, k)
G,H � two graphs
k � number of rounds
Spoiler & Duplicator � two players
11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 38: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/38.jpg)
Ehrenfeucht game
EHR(G,H, k)
G,H � two graphs
k � number of rounds
Spoiler & Duplicator � two players
11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 39: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/39.jpg)
Ehrenfeucht game
EHR(G,H, k)
G,H � two graphs
k � number of rounds
Spoiler & Duplicator � two players
11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 40: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/40.jpg)
Ehrenfeucht game
EHR(G,H, k)
G,H � two graphs
k � number of rounds
Spoiler & Duplicator � two players
11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 41: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/41.jpg)
Ehrenfeucht game
EHR(G,H, k)
G,H � two graphs
k � number of rounds
Spoiler & Duplicator � two players
11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 42: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/42.jpg)
Ehrenfeucht game
EHR(G,H, k)
G,H � two graphs
k � number of rounds
Spoiler & Duplicator � two players
11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 43: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/43.jpg)
Ehrenfeucht game
EHR(G,H, k)
G,H � two graphs
k � number of rounds
Spoiler & Duplicator � two players
11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 44: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/44.jpg)
Ehrenfeucht game
EHR(G,H, k)
G,H � two graphs
k � number of rounds
Spoiler & Duplicator � two players
11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 45: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/45.jpg)
Ehrenfeucht game
EHR(G,H, k)
G,H � two graphs
k � number of rounds
Spoiler & Duplicator � two players
11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 46: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/46.jpg)
Ehrenfeucht game
EHR(G,H, k)
G,H � two graphs
k � number of rounds
Spoiler & Duplicator � two players
11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 47: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/47.jpg)
Ehrenfeucht game
EHR(G,H, k)
G,H � two graphs
k � number of rounds
Spoiler & Duplicator � two players
11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 48: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/48.jpg)
Ehrenfeucht theorem
Theorem [A. Ehrenfeucht, 1960]
Let G,H be two graphs. For any �rst-order property L expressedby formula with quanti�er depth bounded by a number kG ∈ L⇔ H ∈ L if and only if Duplicator has a winning strategy inthe game EHR(G,H, k).
Example: ∃x1 ∃x2 ∃x3 ((x1 ∼ x2) ∧ (x2 ∼ x3) ∧ (x1 ∼ x3)).
12/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 49: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/49.jpg)
Ehrenfeucht theorem
Theorem [A. Ehrenfeucht, 1960]
Let G,H be two graphs. For any �rst-order property L expressedby formula with quanti�er depth bounded by a number kG ∈ L⇔ H ∈ L if and only if Duplicator has a winning strategy inthe game EHR(G,H, k).
Example: ∃x1 ∃x2 ∃x3 ((x1 ∼ x2) ∧ (x2 ∼ x3) ∧ (x1 ∼ x3)).
12/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 50: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/50.jpg)
Ehrenfeucht theorem
Theorem [A. Ehrenfeucht, 1960]
Let G,H be two graphs. For any �rst-order property L expressedby formula with quanti�er depth bounded by a number kG ∈ L⇔ H ∈ L if and only if Duplicator has a winning strategy inthe game EHR(G,H, k).
Example: ∃x1 ∃x2 ∃x3 ((x1 ∼ x2) ∧ (x2 ∼ x3) ∧ (x1 ∼ x3)).
12/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 51: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/51.jpg)
Proofs of k-laws
Corollary
Zero-one law holds if and only if for any k ∈ N almost surelyDuplicator has a winning strategy in the Ehrenfeucht game onk rounds.
Random graph G(n, p) obeys zero-one k-law if and only ifalmost surely Duplicator has a winning strategy in theEhrenfeucht game on k rounds.
13/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 52: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/52.jpg)
Case k = 4, α = 13/14
FISH GRAPH
e = 14, v = 13, ρ = 14/13
G(n, n−13/14) contains a copy of FISH GRAPH with positiveasymptotical probability;
G(n, n−13/14) does not contain a copy of any graph with at most13 vertices and maximal density at least 14/13 with positive
asymptotical probability.
EHR(G,H, 4):
G contains FISH GRAPH;H does not contain a copy of any graph with at most 13vertices and maximal density at least 14/13.
14/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 53: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/53.jpg)
Case k = 4, α = 13/14
FISH GRAPH
e = 14, v = 13, ρ = 14/13
G(n, n−13/14) contains a copy of FISH GRAPH with positiveasymptotical probability;
G(n, n−13/14) does not contain a copy of any graph with at most13 vertices and maximal density at least 14/13 with positive
asymptotical probability.
EHR(G,H, 4):
G contains FISH GRAPH;H does not contain a copy of any graph with at most 13vertices and maximal density at least 14/13.
14/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 54: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/54.jpg)
Case k = 4, α = 13/14
FISH GRAPH
e = 14, v = 13, ρ = 14/13
G(n, n−13/14) contains a copy of FISH GRAPH with positiveasymptotical probability;
G(n, n−13/14) does not contain a copy of any graph with at most13 vertices and maximal density at least 14/13 with positive
asymptotical probability.
EHR(G,H, 4):
G contains FISH GRAPH;H does not contain a copy of any graph with at most 13vertices and maximal density at least 14/13.
14/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 55: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/55.jpg)
Case k = 4, α = 13/14
FISH GRAPH
e = 14, v = 13, ρ = 14/13
G(n, n−13/14) contains a copy of FISH GRAPH with positiveasymptotical probability;
G(n, n−13/14) does not contain a copy of any graph with at most13 vertices and maximal density at least 14/13 with positive
asymptotical probability.
EHR(G,H, 4):
G contains FISH GRAPH;H does not contain a copy of any graph with at most 13vertices and maximal density at least 14/13.
14/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 56: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/56.jpg)
Case k = 4, α = 13/14
FISH GRAPH
e = 14, v = 13, ρ = 14/13
G(n, n−13/14) contains a copy of FISH GRAPH with positiveasymptotical probability;
G(n, n−13/14) does not contain a copy of any graph with at most13 vertices and maximal density at least 14/13 with positive
asymptotical probability.
EHR(G,H, 4):
G contains FISH GRAPH;
H does not contain a copy of any graph with at most 13vertices and maximal density at least 14/13.
14/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 57: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/57.jpg)
Case k = 4, α = 13/14
FISH GRAPH
e = 14, v = 13, ρ = 14/13
G(n, n−13/14) contains a copy of FISH GRAPH with positiveasymptotical probability;
G(n, n−13/14) does not contain a copy of any graph with at most13 vertices and maximal density at least 14/13 with positive
asymptotical probability.
EHR(G,H, 4):
G contains FISH GRAPH;H does not contain a copy of any graph with at most 13vertices and maximal density at least 14/13.
14/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 58: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/58.jpg)
Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 59: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/59.jpg)
Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 60: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/60.jpg)
Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 61: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/61.jpg)
Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 62: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/62.jpg)
Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 63: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/63.jpg)
Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 64: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/64.jpg)
Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 65: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/65.jpg)
Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 66: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/66.jpg)
Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 67: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/67.jpg)
Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 68: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/68.jpg)
Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 69: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/69.jpg)
Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 72: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/72.jpg)
Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Winning strategy of Spoiler
15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
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Open questions
What happens when α ∈(
1k−2 , 1−
12k−1
]?
Spencer and Shelah, 1988: There exists a �rst-order property andan in�nite set S of rational numbers from (0, 1) such that for anyα ∈ S random graph G(n, n−α) follows the property withprobability which doesn't tend to 0 or to 1.
For any �xed k and any ε > 0 in(1− 1
2k−1 + ε, 1)there is only
�nite number of critical points. Does this property holds for(1− 1
2k−1 , 1)? Are there any other such subintervals in(
1k−2 , 1−
12k−1
)?
16/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 80: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/80.jpg)
Open questions
What happens when α ∈(
1k−2 , 1−
12k−1
]?
Spencer and Shelah, 1988: There exists a �rst-order property andan in�nite set S of rational numbers from (0, 1) such that for anyα ∈ S random graph G(n, n−α) follows the property withprobability which doesn't tend to 0 or to 1.
For any �xed k and any ε > 0 in(1− 1
2k−1 + ε, 1)there is only
�nite number of critical points. Does this property holds for(1− 1
2k−1 , 1)? Are there any other such subintervals in(
1k−2 , 1−
12k−1
)?
16/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 81: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/81.jpg)
Open questions
What happens when α ∈(
1k−2 , 1−
12k−1
]?
Spencer and Shelah, 1988: There exists a �rst-order property andan in�nite set S of rational numbers from (0, 1) such that for anyα ∈ S random graph G(n, n−α) follows the property withprobability which doesn't tend to 0 or to 1.
For any �xed k and any ε > 0 in(1− 1
2k−1 + ε, 1)there is only
�nite number of critical points. Does this property holds for(1− 1
2k−1 , 1)? Are there any other such subintervals in(
1k−2 , 1−
12k−1
)?
16/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
![Page 82: 3 zhukovsky](https://reader033.fdocuments.us/reader033/viewer/2022052623/559f63991a28ab4f298b45ed/html5/thumbnails/82.jpg)
Thank you!
Thank you very much for your
attention!
17/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)