3 zhukovsky
0 likes1,548 views
This document discusses zero-one laws for random graphs G(n,p). It defines first-order graph properties and limit probabilities for G(n,p). There is a zero-one law if all first-order properties converge to 0 or 1 as n approaches infinity. The document shows that if p=n-α and α is irrational, G(n,p) obeys the zero-one law. It also discusses bounded quantifier depth properties and critical points where the zero-one law may fail for rational α values.
![Zero-one law
L
the set of all rst-order properties.
Denition
The random graph obeys zero-one law if ∀L ∈ L
lim Pn,p(n) (L) ∈ {0, 1}.
n→∞
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)](https://image.slidesharecdn.com/3zhukovsky-131113053428-phpapp01/85/3-zhukovsky-14-320.jpg)
![Ehrenfeucht theorem
Theorem [A. Ehrenfeucht, 1960]
Let G, H be two graphs. For any rst-order property L expressed
by formula with quantier depth bounded by a number k
G ∈ L ⇔ H ∈ L if and only if Duplicator has a winning strategy in
the 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)](https://image.slidesharecdn.com/3zhukovsky-131113053428-phpapp01/85/3-zhukovsky-48-320.jpg)
![Ehrenfeucht theorem
Theorem [A. Ehrenfeucht, 1960]
Let G, H be two graphs. For any rst-order property L expressed
by formula with quantier depth bounded by a number k
G ∈ L ⇔ H ∈ L if and only if Duplicator has a winning strategy in
the 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)](https://image.slidesharecdn.com/3zhukovsky-131113053428-phpapp01/85/3-zhukovsky-49-320.jpg)
![Ehrenfeucht theorem
Theorem [A. Ehrenfeucht, 1960]
Let G, H be two graphs. For any rst-order property L expressed
by formula with quantier depth bounded by a number k
G ∈ L ⇔ H ∈ L if and only if Duplicator has a winning strategy in
the 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)](https://image.slidesharecdn.com/3zhukovsky-131113053428-phpapp01/85/3-zhukovsky-50-320.jpg)
3 zhukovsky
- 1. 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)
- 2. First-order properties. First-order formulae: relational symbols ∼, =; logical connectivities ¬, ⇒, ⇔, ∨, ∧; variables x, y, x1, ...; quantiers ∀, ∃. 2/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 3. First-order properties. First-order formulae: relational symbols ∼, =; logical connectivities ¬, ⇒, ⇔, ∨, ∧; variables x, y, x1, ...; quantiers ∀, ∃. • property of a graph to be complete ∀x ∀y (¬(x = y) ⇒ (x ∼ y)). 2/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 4. First-order properties. First-order formulae: relational symbols ∼, =; logical connectivities ¬, ⇒, ⇔, ∨, ∧; variables x, y, x1, ...; quantiers ∀, ∃. • 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)). 2/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 5. First-order properties. First-order formulae: relational symbols ∼, =; logical connectivities ¬, ⇒, ⇔, ∨, ∧; variables x, y, x1, ...; quantiers ∀, ∃. • 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)). • 2/17 property of a graph to have chromatic number k Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 6. Limit probabilities. G(n, p): n probability 3/17 number of vertices, p edge appearance Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 7. Limit probabilities. G(n, p): n number of vertices, p edge appearance probability G an arbitrary graph, LG property of containing G. 3/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 8. Limit probabilities. G(n, p): n number of vertices, p edge appearance probability G an arbitrary graph, LG property of containing G. LG is the rst order property. 3/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 9. Limit probabilities. G(n, p): n number of vertices, p edge appearance probability G an arbitrary graph, LG property of containing G. LG is the rst order property. e(H) maximal density ρmax(G) = maxH⊆G{ρ(H)}, ρ(H) = v(H) , p0 = n−1/ρ (G) ; max 3/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 10. Limit probabilities. G(n, p): n number of vertices, p edge appearance probability G an arbitrary graph, LG property of containing G. LG is the rst order property. e(H) maximal density ρmax(G) = maxH⊆G{ρ(H)}, ρ(H) = v(H) , p0 = n−1/ρ (G) ; max p p0 ⇒ lim Pn,p (LG ) = 1, p p0 ⇒ lim Pn,p (LG ) = 0, n→∞ n→∞ p=p0 ⇒ lim Pn,p (LG )∈ {0, 1}. / n→∞ 3/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 11. Limit probabilities. G(n, p): n number of vertices, p edge appearance probability G an arbitrary graph, LG property of containing G. LG is the rst order property. e(H) maximal density ρmax(G) = maxH⊆G{ρ(H)}, ρ(H) = v(H) , p0 = n−1/ρ (G) ; max p p0 ⇒ lim Pn,p (LG ) = 1, p p0 ⇒ lim Pn,p (LG ) = 0, n→∞ n→∞ p=p0 ⇒ lim Pn,p (LG )∈ {0, 1}. / n→∞ If G set of graphs, L = {LG, LG, G ∈ G}, p = n−α , α is irrational 3/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 12. Limit probabilities. G(n, p): n number of vertices, p edge appearance probability G an arbitrary graph, LG property of containing G. LG is the rst order property. e(H) maximal density ρmax(G) = maxH⊆G{ρ(H)}, ρ(H) = v(H) , p0 = n−1/ρ (G) ; max p p0 ⇒ lim Pn,p (LG ) = 1, p p0 ⇒ lim Pn,p (LG ) = 0, n→∞ n→∞ p=p0 ⇒ lim Pn,p (LG )∈ {0, 1}. / n→∞ 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)
- 13. Zero-one law L the set of all rst-order properties. Denition The random graph obeys zero-one law if ∀L ∈ L lim Pn,p(n) (L) ∈ {0, 1}. n→∞ P 4/17 is a set of functions p such that G(n, p) obeys zero-one law. Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 14. Zero-one law L the set of all rst-order properties. Denition The random graph obeys zero-one law if ∀L ∈ L lim Pn,p(n) (L) ∈ {0, 1}. n→∞ 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)
- 15. Quantier depth. Quantier depths of x = y, x ∼ y equal 0. 5/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 16. Quantier depth. Quantier depths of x = y, x ∼ y equal 0. If quantier depth of φ equals k then quantier depth of ¬φ equals k as well. 5/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 17. Quantier depth. Quantier depths of x = y, x ∼ y equal 0. If quantier depth of φ equals k then quantier depth of ¬φ equals k as well. If quantier depth of φ1 equals k1, quantier depth of φ2 equals then quantier depths of φ1 ⇒ φ2, φ1 ⇔ φ2, φ1 ∨ φ2, equal max{k1, k2}. k2 φ1 ∧ φ2 5/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 18. Quantier depth. Quantier depths of x = y, x ∼ y equal 0. If quantier depth of φ equals k then quantier depth of ¬φ equals k as well. If quantier depth of φ1 equals k1, quantier depth of φ2 equals then quantier depths of φ1 ⇒ φ2, φ1 ⇔ φ2, φ1 ∨ φ2, equal max{k1, k2}. k2 φ1 ∧ φ2 If quantier depth of φ(x) equals k, then quantier depths of ∃x φ(x), ∀x φ(x) equal k + 1. 5/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 19. Bounded quantier depth. the class of rst-order properties dened by formulae with quantier depth bounded by k. Random graph G(n, p) obeys zero-one k-law if ∀L ∈ Lk Lk lim Pn,p(n) (L) ∈ {0, 1}. N →∞ the class of all functions p = p(n) such that random graph obeys zero-one k-law. Pk G(n, p) 6/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 20. Bounded quantier depth. the class of rst-order properties dened by formulae with quantier depth bounded by k. Random graph G(n, p) obeys zero-one k-law if ∀L ∈ Lk Lk lim Pn,p(n) (L) ∈ {0, 1}. N →∞ the class of all functions p = p(n) such that random graph obeys zero-one k-law. Pk G(n, p) Example: 6/17 (∀x ∃y (x ∼ y)) ∧ (∀x ∃y ¬(x ∼ y)) ∀x ∃y ∃z ((x ∼ y) ∧ (¬(x ∼ z))) Maksim Zhukovskii k=2 k=3 Critical points in zero-one laws for G(n, p)
- 21. Zero-one k-laws Zhukovskii; 2010 (zero-one Let p=n If 7/17 −α k -law) , k ∈ N, k ≥ 3. 0α 1 k−2 then p ∈ Pk . Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 22. Zero-one k-laws Zhukovskii; 2010 (zero-one Let p=n If −α k -law) , k ∈ N, k ≥ 3. 0α 1 k−2 then p ∈ Pk . Zhukovskii; 2012+ (extension of zero-one Let p=n If 7/17 −α , k ∈ N, k ≥ 4, Q = α=1− 1 , 2k−1 +β {a, b a, b ∈ N, a ≤ 2k−1 } β ∈ (0, ∞) Q Maksim Zhukovskii k -law) then p ∈ Pk . Critical points in zero-one laws for G(n, p)
- 23. Zero-one k-laws Zhukovskii; 2010 (zero-one Let p=n If −α k -law) , k ∈ N, k ≥ 3. 0α 1 k−2 then p ∈ Pk . Zhukovskii; 2012+ (extension of zero-one Let p=n If −α , k ∈ N, k ≥ 4, Q = α=1− 1 , 2k−1 +β 1− 1− ... 1 ,1 2k 1− 1 1 − k−1 2k−1 − 2 3 2 then p ∈ Pk . 1 1 ,1 − k 2k − 1 2 ,1 − Maksim Zhukovskii k -law) a, b ∈ N, a ≤ 2k−1 } β ∈ (0, ∞) Q 1 1 , 1 − k−1 2k−1 + 2k−2 2 + 2k−2 + 1 2k−1 + 7/17 {a, b 1− ... 1 2k−1 + 2k−1 −1 2 ,1 − 1 2k−1 + 2k−2 1 2k−1 + 2k−1 3 . . .? . . . 0, 1 k−2 Critical points in zero-one laws for . G(n, p)
- 24. Zero-one k-laws Zhukovskii; 2013++ (no Let p=n −α k -law) , k ∈ N. If k ≥ 3, α = If 1 k−2 k ≥ 4, then ˜ Q= k−1 2 k−1 − 2 · 1, . . . , 1; 2 −2·1−1·2 ,..., 1; 2 2 2k−1 −2·1−3·2−1·3 , . . . ; 5 ...; α=1− 8/17 p ∈ Pk . / 1 , 2k−1 +β ˜ β∈Q 2 k−1 −2·1−2·2 1 ,..., 3; 3 ...; ...; then k−1 −2·1−3·2 ,..., 1; 4 4 2k−1 −2·1−3·2−4·3 ,...; 8 2 ... p ∈ Pk . / Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 25. Critical points 1−α: 0→1 9/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 26. Critical points 1−α: 0→1 0 9/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 27. Critical points 1−α: 0→1 0 9/17 1 2k Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 28. Critical points 1−α: 0→1 0 9/17 1 2k 1 2k −1 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 29. Critical points 1−α: 0→1 0 9/17 1 2k 1 2k −1 1 2k −2 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 30. Critical points 1−α: 0→1 0 9/17 1 2k 1 2k −1 1 2k −2 1 2k −3 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 31. Critical points 1−α: 0→1 0 9/17 1 2k 1 2k −1 1 2k −2 1 2k −3 ... Maksim Zhukovskii 1 2k −2k−2 Critical points in zero-one laws for G(n, p)
- 32. Critical points 1−α: 0→1 0 9/17 1 2k 1 2k −1 1 2k −2 1 2k −3 ... Maksim Zhukovskii 1 2k −2k−2 ... Critical points in zero-one laws for G(n, p)
- 33. Critical points 1−α: 0→1 0 1 2k 1 2k −1 Large gap: 9/17 1 2k −2 1 2k −3 ... 1 2k −2k−2 ... ? [1/2k−1 , 1 − 1/(k − 2)) Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 34. Critical points 1−α: 0→1 0 1 2k 1 2k −1 Large gap: 9/17 1 2k −2 1 2k −3 ... 1 2k −2k−2 ... ? 1− 1 k−2 [1/2k−1 , 1 − 1/(k − 2)) Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 35. Critical points 1−α: 0→1 0 Large gap: 9/17 1 2k −2 1 2k −3 ... 1 2k −2k−2 ... ? 1− 1 k−2 [1/2k−1 , 1 − 1/(k − 2)) Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 36. Intervals 10/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 37. Ehrenfeucht game EHR(G, H, k) two graphs k number of rounds Spoiler Duplicator two players G, H 11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 38. Ehrenfeucht game EHR(G, H, k) two graphs k number of rounds Spoiler Duplicator two players G, H 11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 39. Ehrenfeucht game EHR(G, H, k) two graphs k number of rounds Spoiler Duplicator two players G, H 11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 40. Ehrenfeucht game EHR(G, H, k) two graphs k number of rounds Spoiler Duplicator two players G, H 11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 41. Ehrenfeucht game EHR(G, H, k) two graphs k number of rounds Spoiler Duplicator two players G, H 11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 42. Ehrenfeucht game EHR(G, H, k) two graphs k number of rounds Spoiler Duplicator two players G, H 11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 43. Ehrenfeucht game EHR(G, H, k) two graphs k number of rounds Spoiler Duplicator two players G, H 11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 44. Ehrenfeucht game EHR(G, H, k) two graphs k number of rounds Spoiler Duplicator two players G, H 11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 45. Ehrenfeucht game EHR(G, H, k) two graphs k number of rounds Spoiler Duplicator two players G, H 11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 46. Ehrenfeucht game EHR(G, H, k) two graphs k number of rounds Spoiler Duplicator two players G, H 11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 47. Ehrenfeucht game EHR(G, H, k) two graphs k number of rounds Spoiler Duplicator two players G, H 11/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 48. Ehrenfeucht theorem Theorem [A. Ehrenfeucht, 1960] Let G, H be two graphs. For any rst-order property L expressed by formula with quantier depth bounded by a number k G ∈ L ⇔ H ∈ L if and only if Duplicator has a winning strategy in the 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)
- 49. Ehrenfeucht theorem Theorem [A. Ehrenfeucht, 1960] Let G, H be two graphs. For any rst-order property L expressed by formula with quantier depth bounded by a number k G ∈ L ⇔ H ∈ L if and only if Duplicator has a winning strategy in the 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)
- 50. Ehrenfeucht theorem Theorem [A. Ehrenfeucht, 1960] Let G, H be two graphs. For any rst-order property L expressed by formula with quantier depth bounded by a number k G ∈ L ⇔ H ∈ L if and only if Duplicator has a winning strategy in the 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)
- 51. Proofs of k-laws Corollary Zero-one law holds if and only if for any k ∈ N almost surely Duplicator has a winning strategy in the Ehrenfeucht game on k rounds. Random graph G(n, p) obeys zero-one k-law if and only if almost surely Duplicator has a winning strategy in the Ehrenfeucht game on k rounds. 13/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 52. Case k = 4, α = 13/14 14/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 53. Case k = 4, α = 13/14 FISH GRAPH , , e = 14 v = 13 ρ = 14/13 14/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 54. Case k = 4, α = 13/14 FISH GRAPH , , G(n, n−13/14 ) contains a copy of FISH GRAPH with positive asymptotical probability; G(n, n−13/14 ) does not contain a copy of any graph with at most 13 vertices and maximal density at least 14/13 with positive asymptotical probability. e = 14 v = 13 ρ = 14/13 14/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 55. Case k = 4, α = 13/14 FISH GRAPH , , G(n, n−13/14 ) contains a copy of FISH GRAPH with positive asymptotical probability; G(n, n−13/14 ) does not contain a copy of any graph with at most 13 vertices and maximal density at least 14/13 with positive asymptotical probability. EHR(G, H, 4): e = 14 v = 13 ρ = 14/13 14/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 56. Case k = 4, α = 13/14 FISH GRAPH , , G(n, n−13/14 ) contains a copy of FISH GRAPH with positive asymptotical probability; G(n, n−13/14 ) does not contain a copy of any graph with at most 13 vertices and maximal density at least 14/13 with positive asymptotical probability. EHR(G, H, 4): G contains FISH GRAPH; e = 14 v = 13 ρ = 14/13 14/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 57. Case k = 4, α = 13/14 FISH GRAPH , , G(n, n−13/14 ) contains a copy of FISH GRAPH with positive asymptotical probability; G(n, n−13/14 ) does not contain a copy of any graph with at most 13 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 13 vertices and maximal density at least 14/13. e = 14 v = 13 ρ = 14/13 14/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 58. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 59. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 60. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 61. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 62. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 63. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 64. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 65. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 66. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 67. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 68. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 69. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 70. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 71. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 72. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 73. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 74. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 75. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 76. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 77. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 78. Winning strategy of Spoiler 15/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 79. Open questions What happens when α ∈ 16/17 1 k−2 , 1 Maksim Zhukovskii − 1 2k−1 ? Critical points in zero-one laws for G(n, p)
- 80. Open questions What happens when α ∈ 1 k−2 , 1 − 1 2k−1 ? Spencer and Shelah, 1988: There exists a rst-order property and an innite set S of rational numbers from (0, 1) such that for any α ∈ S random graph G(n, n−α ) follows the property with probability which doesn't tend to 0 or to 1. 16/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 81. Open questions What happens when α ∈ 1 k−2 , 1 − 1 2k−1 ? Spencer and Shelah, 1988: There exists a rst-order property and an innite set S of rational numbers from (0, 1) such that for any α ∈ S random graph G(n, n−α ) follows the property with probability which doesn't tend to 0 or to 1. For any xed k and any ε 0 in 1 − 2 1 + ε, 1 there is only nite number of critical points. Does this property holds for 1 − 2 1 , 1 ? Are there any other such subintervals in 1 1 ? k−2 , 1 − 2 k−1 k−1 k−1 16/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)
- 82. Thank you! Thank you very much for your attention! 17/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)