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.

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Quantier depth.
Quantier depths of x = y, x ∼ y equal 0.
Maksim Zhukovskii Zero-one k-laws for G(n, n−α)

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Critical points
α : 0 → 1
Maksim Zhukovskii Zero-one k-laws for G(n, n−α)

28. Critical points
α : 0 → 1
1
k−2
Maksim Zhukovskii Zero-one k-laws for G(n, n−α)

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. 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. 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. 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. 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. 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. 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. 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. 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. Intervals
Maksim Zhukovskii Zero-one k-laws for G(n, n−α)

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Thank you!
Thank you very much for your
attention!
Maksim Zhukovskii Zero-one k-laws for G(n, n−α)