Upcoming SlideShare
Loading in …5
×

# Maksim Zhukovskii – Zero-one k-laws for G(n,n−α)

392 views

Published on

We study asymptotical behavior of the probabilities of first-order properties for Erdős-Rényi random graphs G(n,p(n)) with p(n)=n-α, α ∈ (0,1). The following zero-one law was proved in 1988 by S. Shelah and J.H. Spencer [1]: if α is irrational then for any first-order property L either the random graph satisfies the property L asymptotically almost surely or it doesn't satisfy (in such cases the random graph is said to obey zero-one law. When α ∈ (0,1) is rational the zero-one law for these graphs doesn't hold.

Let k be a positive integer. Denote by Lk the class of the first-order properties of graphs defined by formulae with quantifier depth bounded by the number k (the sentences are of a finite length). Let us say that the random graph obeys zero-one k-law, if for any first-order property L ∈ Lk either the random graph satisfies the property L almost surely or it doesn't satisfy. Since 2010 we prove several zero-one \$k\$-laws for rational α from Ik=(0, 1/(k-2)] ∪ [1-1/(2k-1), 1). For some points from Ik we disprove the law. In particular, for α ∈ (0, 1/(k-2)) ∪ (1-1/2k-2, 1) zero-one k-law holds. If α ∈ {1/(k-2), 1-1/(2k-2)}, then zero-one law does not hold (in such cases we call the number α k-critical).

We also disprove the law for some α ∈ [2/(k-1), k/(k+1)]. From our results it follows that zero-one 3-law holds for any α ∈ (0,1). Therefore, there are no 3-critical points in (0,1). Zero-one 4-law holds when α ∈ (0,1/2) ∪ (13/14,1). Numbers 1/2 and 13/14 are 4-critical. Moreover, we know some rational 4-critical and not 4-critical numbers in [7/8,13/14). The number 2/3 is 4-critical. Recently we obtain new results concerning zero-one 4-laws for the neighborhood of the number 2/3.

References

[1] S. Shelah, J.H. Spencer, Zero-one laws for sparse random graphs, J. Amer. Math. Soc.

1: 97–115, 1988.

Published in: Science, Technology
0 Comments
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
Your message goes here
• Be the first to comment

• Be the first to like this

No Downloads
Views
Total views
392
On SlideShare
0
From Embeds
0
Number of Embeds
37
Actions
Shares
0
Downloads
7
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

### Maksim Zhukovskii – Zero-one k-laws for G(n,n−α)

1. 1. Zero-one k-laws for G(n, n−α ) Maksim Zhukovskii MSU, MIPT, Yandex Workshop on Extremal Graph Theory 6 June 2014 Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
2. 2. First-order properties. First-order formulae: relational symbols ∼, =; logical connectivities ¬, ⇒, ⇔, ∨, ∧; variables x, y, x1, ...; quantiers ∀, ∃. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
3. 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)). Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
4. 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)). Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
5. 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)). • property of a graph to have chromatic number k Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
6. 6. Limit probabilities. G(n, p): n number of vertices, p edge appearance probability Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
7. 7. Limit probabilities. G(n, p): n number of vertices, p edge appearance probability G an arbitrary graph, LG property of containing G. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
8. 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. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
9. 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. maximal density ρmax (G) = maxH⊆G{ρ(H)}, ρ(H) = e(H) v(H) , p0 = n−1/ρmax(G) ; Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
10. 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. maximal density ρmax (G) = maxH⊆G{ρ(H)}, ρ(H) = e(H) v(H) , p0 = n−1/ρmax(G) ; p p0 ⇒ lim n→∞ Pn,p(LG) = 1, p p0 ⇒ lim n→∞ Pn,p(LG) = 0, p=p0 ⇒ lim n→∞ Pn,p(LG)/∈ {0, 1}. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
11. 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. maximal density ρmax (G) = maxH⊆G{ρ(H)}, ρ(H) = e(H) v(H) , p0 = n−1/ρmax(G) ; p p0 ⇒ lim n→∞ Pn,p(LG) = 1, p p0 ⇒ lim n→∞ Pn,p(LG) = 0, p=p0 ⇒ lim n→∞ Pn,p(LG)/∈ {0, 1}. If G set of graphs, L = {LG, LG, G ∈ G}, p = n−α , α is irrational Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
12. 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. maximal density ρmax (G) = maxH⊆G{ρ(H)}, ρ(H) = e(H) v(H) , p0 = n−1/ρmax(G) ; p p0 ⇒ lim n→∞ Pn,p(LG) = 1, p p0 ⇒ lim n→∞ Pn,p(LG) = 0, p=p0 ⇒ lim n→∞ 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}. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
13. 13. Zero-one law L the set of all rst-order properties. Denition The random graph obeys zero-one law if ∀L ∈ L lim n→∞ Pn,p(n)(L) ∈ {0, 1}. P is a set of functions p such that G(n, p) obeys zero-one law. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
14. 14. Zero-one law L the set of all rst-order properties. Denition The random graph obeys zero-one law if ∀L ∈ L lim n→∞ 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. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
15. 15. Quantier depth. Quantier depths of x = y, x ∼ y equal 0. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
16. 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. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
17. 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 k2 then quantier depths of φ1 ⇒ φ2, φ1 ⇔ φ2, φ1 ∨ φ2, φ1 ∧ φ2 equal max{k1, k2}. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
18. 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 k2 then quantier depths of φ1 ⇒ φ2, φ1 ⇔ φ2, φ1 ∨ φ2, φ1 ∧ φ2 equal max{k1, k2}. If quantier depth of φ(x) equals k, then quantier depths of ∃x φ(x), ∀x φ(x) equal k + 1. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
19. 19. Bounded quantier depth. Lk 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 lim n→∞ Pn,p(n)(L) ∈ {0, 1}. Pk the class of all functions p = p(n) such that random graph G(n, p) obeys zero-one k-law. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
20. 20. Bounded quantier depth. Lk 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 lim n→∞ Pn,p(n)(L) ∈ {0, 1}. Pk the class of all functions p = p(n) such that random graph G(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 Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
21. 21. Examples Simple case: α ∈ (0, 1 k−1 ), zero-one k-law holds. The proof is very simple. Method of proof from Theorem of Glebskii et el.: Theorem [Y.V. Glebskii, D.I. Kogan, M.I. Liagonkii, V.A. Talanov, 1969; R.Fagin, 1976] Let for any β 0 min{p, 1 − p}nβ → ∞, n → ∞. Then p ∈ P. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
22. 22. Examples Simple case: α ∈ (0, 1 k−1 ), zero-one k-law holds. The proof is very simple. Method of proof from Theorem of Glebskii et el.: Theorem [Y.V. Glebskii, D.I. Kogan, M.I. Liagonkii, V.A. Talanov, 1969; R.Fagin, 1976] Let for any β 0 min{p, 1 − p}nβ → ∞, n → ∞. Then p ∈ P. What happens when α ≥ 1 k−1 ? The most dense graph is Kk. Therefore, the rst α such that zero-one k-law does not hold for LG is 2 k−1 . Does zero-one k-law hold when α ∈ [ 1 k−1 , 2 k−1 )? Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
23. 23. Zero-one k-laws Zhukovskii; 2010 (zero-one k-law) Let p = n−α , k ∈ N, k ≥ 3. If 0 α 1 k−2 then p ∈ Pk. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
24. 24. Zero-one k-laws Zhukovskii; 2010 (zero-one k-law) Let p = n−α , k ∈ N, k ≥ 3. If 0 α 1 k−2 then p ∈ Pk. Zhukovskii; 2012+ (extension of zero-one k-law) Let p = n−α , k ∈ N, k ≥ 4, Q = {a b , a, b ∈ N, a ≤ 2k−1 } If α = 1 − 1 2k−1+β , β ∈ (0, ∞) Q then p ∈ Pk. If α = 1 − 1 2k−1+β , β ∈ {2k−1 − 1, 2k−1 } then p ∈ Pk. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
25. 25. Zero-one k-laws Zhukovskii; 2010 (zero-one k-law) Let p = n−α , k ∈ N, k ≥ 3. If 0 α 1 k−2 then p ∈ Pk. Zhukovskii; 2012+ (extension of zero-one k-law) Let p = n−α , k ∈ N, k ≥ 4, Q = {a b , a, b ∈ N, a ≤ 2k−1 } If α = 1 − 1 2k−1+β , β ∈ (0, ∞) Q then p ∈ Pk. If α = 1 − 1 2k−1+β , β ∈ {2k−1 − 1, 2k−1 } then p ∈ Pk. 1 − 1 2k − 2 , 1 1 − 1 2k − 3 , 1 − 1 2k − 2 . . . 1 − 1 2k−1 + 2k−2 , 1 − 1 2k−1 + 2k−2 + 1 1 − 1 2k−1 + 2k−1−1 2 , 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 . Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
26. 26. Zero-one k-laws Zhukovskii; 2013++ (no k-law) Let p = n−α , k ∈ N. If k ≥ 3, α = 1 k−2 then p /∈ Pk. If k ≥ 4, ˜Q =    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 −2·1−3·2−1·3 5 , . . . ; . . . ; 2k−1 −2·1−3·2−4·3 8 , . . . ; . . . ; . . . ; . . .    α = 1 − 1 2k−1+β , β ∈ ˜Q then p /∈ Pk. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
27. 27. Critical points α : 0 → 1 Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
28. 28. Critical points α : 0 → 1 1 k−2 Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
29. 29. Critical points α : 0 → 1 1 k−2 ? Large gap: (1/(k − 2), 1 − 1/2k−1 ) Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
30. 30. Critical points α : 0 → 1 1 k−2 ? 1 − 1 2k−1 Large gap: (1/(k − 2), 1 − 1/2k−1 ) Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
31. 31. Critical points α : 0 → 1 1 k−2 ? 1 − 1 2k−1 . . . Large gap: (1/(k − 2), 1 − 1/2k−1 ) Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
32. 32. Critical points α : 0 → 1 1 k−2 ? 1 − 1 2k−1 . . . 1 2k−2k−2 Large gap: (1/(k − 2), 1 − 1/2k−1 ) Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
33. 33. Critical points α : 0 → 1 1 k−2 ? 1 − 1 2k−1 . . . 1 2k−2k−2 . . . 1 2k−3 Large gap: (1/(k − 2), 1 − 1/2k−1 ) Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
34. 34. Critical points α : 0 → 1 1 k−2 ? 1 − 1 2k−1 . . . 1 2k−2k−2 . . . 1 2k−3 1 2k−2 Large gap: (1/(k − 2), 1 − 1/2k−1 ) Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
35. 35. Critical points α : 0 → 1 1 k−2 ? 1 − 1 2k−1 . . . 1 2k−2k−2 . . . 1 2k−3 1 2k−2 0 Large gap: (1/(k − 2), 1 − 1/2k−1 ) Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
36. 36. Cases k = 3, k = 4 k = 3 For any α ∈ (0, 1) p ∈ P3. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
37. 37. Cases k = 3, k = 4 k = 3 For any α ∈ (0, 1) p ∈ P3. k = 4 For any α ∈ (0, 1/2) p ∈ P4. For any α ∈ (13/14, 1) p ∈ P4. Some results for α ∈ (7/8, 13/14) . . . If α ∈ {1/2, 2/3, 3/4, 5/6, 7/8, 9/10, 10/11, 11/12, 12/13, 13/14}, then p /∈ P4. For any α ∈ (184/277, 2/3) p ∈ P4. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
38. 38. Intervals Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
39. 39. k = 4 α = 2/3, L the property of containing K4 α = 3/4, ϕ = ∃x1 ((∃x2∃x3∃x4 ((x3 ∼ x1)∧(x2 ∼ x1)∧(x2 ∼ x3)))∧ (∃x2∃x3 ((x1 ∼ x2) ∧ (x1 ∼ x3) ∧ (x2 ∼ x3)))∧ (∀x4 ((x1 ∼ x4) → ((¬(x2 ∼ x4)) ∧ (¬(x3 ∼ x4)))))) Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
40. 40. k = 4 α = 4/5, L the property of containing K4 without one edge α = 5/6, ϕ = ∃x1∃x2 ((x1 ∼ x2) ∧ (∃x3 ((x3 ∼ x1) ∧ (x3 ∼ x2)))∧ (∃x3∃x4 ((x1 ∼ x3) ∧ (x3 ∼ x4) ∧ (x4 ∼ x2)))) Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
41. 41. Ehrenfeucht game EHR(G, H, k) G, H two graphs k number of rounds Spoiler Duplicator two players Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
42. 42. Ehrenfeucht game EHR(G, H, k) G, H two graphs k number of rounds Spoiler Duplicator two players Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
43. 43. Ehrenfeucht game EHR(G, H, k) G, H two graphs k number of rounds Spoiler Duplicator two players Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
44. 44. Ehrenfeucht game EHR(G, H, k) G, H two graphs k number of rounds Spoiler Duplicator two players Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
45. 45. Ehrenfeucht game EHR(G, H, k) G, H two graphs k number of rounds Spoiler Duplicator two players Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
46. 46. Ehrenfeucht game EHR(G, H, k) G, H two graphs k number of rounds Spoiler Duplicator two players Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
47. 47. Ehrenfeucht game EHR(G, H, k) G, H two graphs k number of rounds Spoiler Duplicator two players Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
48. 48. Ehrenfeucht game EHR(G, H, k) G, H two graphs k number of rounds Spoiler Duplicator two players Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
49. 49. Ehrenfeucht game EHR(G, H, k) G, H two graphs k number of rounds Spoiler Duplicator two players Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
50. 50. Ehrenfeucht game EHR(G, H, k) G, H two graphs k number of rounds Spoiler Duplicator two players Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
51. 51. Ehrenfeucht game EHR(G, H, k) G, H two graphs k number of rounds Spoiler Duplicator two players Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
52. 52. 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)). Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
53. 53. 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)). Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
54. 54. 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)). Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
55. 55. 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. Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
56. 56. Open questions What happens when α ∈ 1 k−2 , 1 − 1 2k−1 ? Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
57. 57. Open questions What happens when α ∈ 1 k−2 , 1 − 1 2k−1 ? Spencer and Shelah: 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 − 1 2k−1 + ε, 1 there is only nite number of critical points. Does this property holds for 1 − 1 2k−1 , 1 ? Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
58. 58. Open questions What happens when α ∈ 1 k−2 , 1 − 1 2k−1 ? Spencer and Shelah: 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 − 1 2k−1 + ε, 1 there is only nite number of critical points. Does this property holds for 1 − 1 2k−1 , 1 ? What is the maximal k such that there is nite number of critical points? Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
59. 59. Open questions What happens when α ∈ 1 k−2 , 1 − 1 2k−1 ? Spencer and Shelah: 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 − 1 2k−1 + ε, 1 there is only nite number of critical points. Does this property holds for 1 − 1 2k−1 , 1 ? What is the maximal k such that there is nite number of critical points? k = 4, k = 5, . . . Maksim Zhukovskii Zero-one k-laws for G(n, n−α)
60. 60. Thank you! Thank you very much for your attention! Maksim Zhukovskii Zero-one k-laws for G(n, n−α)