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 and extended zero-one
k-laws for random distance graphs
Popova Svetlana
vomonosov wos™ow ƒt—te …niversity
‡orkshop on ixtrem—l qr—ph „heory
wos™owD tune TD PHIR
1/22

2. he(nitionsX ird¥osE‚¡enyi r—ndom gr—ph G(n, p) —nd r—ndom gr—ph G(Gn, p)
he(nitionF ird¥osE‚¡enyi r—ndom gr—ph G(n, p) is — r—ndom element
with v—lues in Ωn —nd distri˜ution Pn,p on FnD where
Ωn = {(V = {1, ..., n}, E)}, Fn = 2Ωn
,
Pn,p(G) = p|E|
(1 − p)C2
n−|E|
.
2/22

3. he(nitionsX ird¥osE‚¡enyi r—ndom gr—ph G(n, p) —nd r—ndom gr—ph G(Gn, p)
he(nitionF ird¥osE‚¡enyi r—ndom gr—ph G(n, p) is — r—ndom element
with v—lues in Ωn —nd distri˜ution Pn,p on FnD where
Ωn = {(V = {1, ..., n}, E)}, Fn = 2Ωn
,
Pn,p(G) = p|E|
(1 − p)C2
n−|E|
.
he(nitionF vet Gn ˜e — sequen™e of gr—phs Gn = (Vn, En)F
‚—ndom gr—ph G(Gn, p) is — r—ndom element with v—lues in ΩGn
—nd distri˜ution PGn,p on FGn D where
ΩGn = {G = (V, E) : V = Vn, E ⊆ En},
FGn = 2ΩGn , PGn,p(G) = p|E|
(1 − p)|En|−|E|
.
2/22

4. he(nitionsX (rstEorder properties —nd zeroEone l—w
he(nitionF pirstEorder properties of gr—phs —re de(ned ˜y
(rstEorder formul—eD whi™h —re ˜uilt of
predi™—te sym˜ols ∼, =
logi™—l ™onne™tivities ¬, ⇒, ⇔, ∨, ∧
v—ri—˜les x, y, . . .
qu—nti(ers ∀, ∃
3/22

5. he(nitionsX (rstEorder properties —nd zeroEone l—w
he(nitionF pirstEorder properties of gr—phs —re de(ned ˜y
(rstEorder formul—eD whi™h —re ˜uilt of
predi™—te sym˜ols ∼, =
logi™—l ™onne™tivities ¬, ⇒, ⇔, ∨, ∧
v—ri—˜les x, y, . . .
qu—nti(ers ∀, ∃
he(nitionF „he r—ndom gr—ph G(n, p) is s—id to follow zeroEone l—w
if for —ny (rstEorder property L either
lim
n→∞
Pn,p(L) = 0
or
lim
n→∞
Pn,p(L) = 1.
3/22

6. he(nitionsX zeroEone kEl—w
he(nitionF „he r—ndom gr—ph G(n, p) is s—id to follow zeroEone
kEl—w if for —ny property L de(ned ˜y — (rstEorder formul— with
qu—nti(er depth —t most k either
lim
n→∞
Pn,p(L) = 0
or
lim
n→∞
Pn,p(L) = 1.
4/22

7. eroEone l—w for ird¥osE‚¡enyi r—ndom gr—ph G(n, p)
„heorem@qle˜ski et —lFD IWTWY p—ginD IWUTA
Let a function p = p(n) satisfy the property
∀β > 0 min(p, 1 − p)nβ
→ ∞ when n → ∞.
Then the random graph G(n, p) follows the zero-one law.
5/22

8. eroEone l—w for ird¥osE‚¡enyi r—ndom gr—ph G(n, p)
„heorem@qle˜ski et —lFD IWTWY p—ginD IWUTA
Let a function p = p(n) satisfy the property
∀β > 0 min(p, 1 − p)nβ
→ ∞ when n → ∞.
Then the random graph G(n, p) follows the zero-one law.
„heorem@ƒhel—hD ƒpen™erD IWVVA
Let p(n) = n−β and β be an irrational number, 0 < β < 1.
Then the random graph G(n, p) follows the zero-one law.
5/22

9. ‚—ndom dist—n™e gr—ph
‚—ndom dist—n™e gr—ph G(Gdist
n , p)
Gdist
n = (V dist
n , Edist
n )
a = a(n), c = c(n)
V dist
n = v = (v1
, . . . , vn
) : vi
∈ {0, 1},
n
i=1
vi
= a
Edist
n = {{u, v} ∈ V dist
n × V dist
n : (u, v) = c}
6/22

10. eroEone l—w for r—ndom dist—n™e gr—ph
vet — fun™tion p = p(n) s—tisfy the property
∀β > 0 min(p, 1 − p)|V dist
n |β
→ ∞ when n → ∞.
7/22

11. eroEone l—w for r—ndom dist—n™e gr—ph
vet — fun™tion p = p(n) s—tisfy the property
∀β > 0 min(p, 1 − p)|V dist
n |β
→ ∞ when n → ∞.
„heorem
Let a(n) = αn, c(n) = α2n, α ∈ Q, 0 < α < 1. Then
the random graph G(Gdist
n , p) doesn't follow the zero-one law, but
there exists a subsequence G(Gdist
ni
, p) following the zero-one law.
7/22

12. uestions
‡hen does — given su˜sequen™e G(Gdist
ni
, p) follow zeroEone
l—wc
8/22

13. uestions
‡hen does — given su˜sequen™e G(Gdist
ni
, p) follow zeroEone
l—wc
hoes there exist — (rstEorder property L —nd — su˜sequen™e
G(Gdist
ni
, p) su™h th—t
lim
i→∞
PGdist
ni
,p(L) ∈ (0, 1)
8/22

14. uestions
‡hen does — given su˜sequen™e G(Gdist
ni
, p) follow zeroEone
l—wc
hoes there exist — (rstEorder property L —nd — su˜sequen™e
G(Gdist
ni
, p) su™h th—t
lim
i→∞
PGdist
ni
,p(L) ∈ (0, 1)
‡h—t limiting pro˜—˜ilities PGdist
ni
,p(L) ™—n we getc
8/22

15. ixtended zeroEone kEl—w
he(nitionF „he r—ndom gr—ph G(Gn, p) is s—id to follow extended
zeroEone kEl—w if for every property L de(ned ˜y — (rstEorder
formul— with qu—nti(er depth —t most k —ny p—rti—l limit of the
sequen™e PGn,p(L) equ—ls either 0 or 1F
9/22

16. ixtended zeroEone kEl—w
he(nitionF „he r—ndom gr—ph G(Gn, p) is s—id to follow extended
zeroEone kEl—w if for every property L de(ned ˜y — (rstEorder
formul— with qu—nti(er depth —t most k —ny p—rti—l limit of the
sequen™e PGn,p(L) equ—ls either 0 or 1F
qo—lF pind ™onditions on the sequen™e G(Gdist
ni
, p) under whi™h one
of the following t—kes pl—™eX
zeroEone kEl—w holds
zeroEone kEl—w doesn9t holdD ˜ut extended zeroEone kEl—w holds
extended zeroEone kEl—w doesn9t hold
9/22

17. ihrenfeu™ht g—me EHR(G, H, k)
EHR(G, H, k)
qr—phs G, HD num˜er of rounds k
„wo pl—yers ƒpoiler —nd hupli™—tor
iEth roundX
ƒpoiler ™hooses — vertex either from G or from H
hupli™—tor ™hooses — vertex of the other gr—ph
vet x1, . . . , xkD y1, . . . , yk ˜e verti™es ™hosen from gr—phs G —nd
H respe™tivelyF
hupli™—tor wins if —nd only if G|{x1,...,xk}
∼= H|{y1,...,yk}F
10/22

18. ihrenfeu™ht g—me EHR(G, H, k)
EHR(G, H, k)
qr—phs G, HD num˜er of rounds k
„wo pl—yers ƒpoiler —nd hupli™—tor
iEth roundX
ƒpoiler ™hooses — vertex either from G or from H
hupli™—tor ™hooses — vertex of the other gr—ph
vet x1, . . . , xkD y1, . . . , yk ˜e verti™es ™hosen from gr—phs G —nd
H respe™tivelyF
hupli™—tor wins if —nd only if G|{x1,...,xk}
∼= H|{y1,...,yk}F
„heorem
The random graph G(Gn, p) follows zero-one k-law if and only if
P(Duplicator wins the game EHR(G(Gn, p), G(Gm, p), k)) → 1
as n, m → ∞.
10/22

19. pull level extension property
he(nitionF „he gr—ph G = (V, E) is s—id to s—tisfy full level t
extension property if for —ny verti™es v1, . . . , vl, u1, . . . , ur
(l + r ≤ t) there exists — vertex v —dj—™ent to v1, . . . , vl —nd
nonE—dj—™ent to u1, . . . , urF
11/22

20. pull level extension property
he(nitionF „he gr—ph G = (V, E) is s—id to s—tisfy full level t
extension property if for —ny verti™es v1, . . . , vl, u1, . . . , ur
(l + r ≤ t) there exists — vertex v —dj—™ent to v1, . . . , vl —nd
nonE—dj—™ent to u1, . . . , urF
€roposition
Let G(Gn, p) satisfy full level (k − 1) extension property
asymptotically almost surely. Then the random graph G(Gn, p)
follows zero-one k-law.
11/22

21. pull level extension property
he(nitionF „he gr—ph G = (V, E) is s—id to s—tisfy full level t
extension property if for —ny verti™es v1, . . . , vl, u1, . . . , ur
(l + r ≤ t) there exists — vertex v —dj—™ent to v1, . . . , vl —nd
nonE—dj—™ent to u1, . . . , urF
€roposition
Let G(Gn, p) satisfy full level (k − 1) extension property
asymptotically almost surely. Then the random graph G(Gn, p)
follows zero-one k-law.
goroll—ry
Let G(Gn, p) satisfy full level t extension property a.a.s for every
t ∈ N. Then the random graph G(Gn, p) follows the zero-one law.
11/22

22. pull level extension property for r—ndom dist—n™e gr—ph
€roposition
Let a(n) = αn, α ∈ Q, 0 < α < 1. Then G(Gdist
ni
, p) satises
full level t extension property a.a.s for every t ∈ N if and only if
c = α2n and ∀m ∈ N m|ni for suciently large i.
12/22

23. pull level extension property for r—ndom dist—n™e gr—ph
€roposition
Let a(n) = αn, α ∈ Q, 0 α 1. Then G(Gdist
ni
, p) satises
full level t extension property a.a.s for every t ∈ N if and only if
c = α2n and ∀m ∈ N m|ni for suciently large i.
€roposition
Let a(n) = αn, c = α2n, α ∈ Q, 0 α 1, t ≤ 5. Then
G(Gdist
ni
, p) satises full level t extension property a.a.s if and only if
Dt|a(ni) − c(ni) for suciently large i, where
D2 = 1, D3 = 2, D4 = 6, D5 = 60.
12/22

24. eroEone kEl—ws for r—ndom dist—n™e gr—ph
xot—tionF a = αn, c = α2n, α = s/q, (s, q) = 1.
13/22

25. eroEone kEl—ws for r—ndom dist—n™e gr—ph
xot—tionF a = αn, c = α2n, α = s/q, (s, q) = 1.
„heorem @zeroEone 4El—wA
The random graph G(Gdist
n , p) follows extended zero-one 4-law.
The sequence G(Gdist
ni
, p) follows zero-one 4-law if and only if
∃i0 such that all the numbers a(ni) − c(ni) for i i0 have the
same parity.
13/22

26. eroEone kEl—ws for r—ndom dist—n™e gr—ph
xot—tionF a = αn, c = α2n, α = s/q, (s, q) = 1.
„heorem @zeroEone 4El—wA
The random graph G(Gdist
n , p) follows extended zero-one 4-law.
The sequence G(Gdist
ni
, p) follows zero-one 4-law if and only if
∃i0 such that all the numbers a(ni) − c(ni) for i i0 have the
same parity.
„heorem @zeroEone 5El—wA
Let a sequence {ni} be such that a(ni) − c(ni) are even for
suciently large i. Then
G(Gdist
ni
, p) follows extended zero-one 5-law,
G(Gdist
ni
, p) follows zero-one 5-law if and only if ∃i0 such that
either ∀i i0 3|a(ni) − c(ni) or ∀i i0 3 a(ni) − c(ni).
13/22

27. eroEone kEl—ws for r—ndom dist—n™e gr—ph
„heorem @zeroEone 6El—wA
Let q = 5 and a sequence {ni} be such that a(ni) − c(ni) are
divisible by 12 for suciently large i. Then
G(Gdist
ni
, p) follows extended zero-one 6-law,
G(Gdist
ni
, p) follows zero-one 6-law if and only if ∃i0 such that
either ∀i i0 5|a(ni) − c(ni) or ∀i i0 5 a(ni) − c(ni).
14/22

28. hisproof of extended zeroEone l—ws for r—ndom dist—n™e gr—ph
∀β 0 min(p, 1 − p)|V dist
n |β
→ ∞ —s n → ∞. (∗)
„heorem @disproof of extended zeroEone 6El—wA
Let one of the following two cases take place:
q = 5 and a sequence {ni} is such that a(ni) − c(ni) are not
divisible by 5 for suciently large i,
α = 1
2 and a sequence {ni} is such that a(ni) − c(ni) are not
divisible by 4 for suciently large i.
Then there exists a function p(n) satisfying (∗) such that
G(Gdist
ni
, p) doesn't follow extended zero-one 6-law.
15/22

29. hisproof of extended zeroEone l—ws for r—ndom dist—n™e gr—ph
„heorem @disproof of extended zeroEone l—wA
Let q be even, α ∈ (1
4, 3
4) and a sequence {ni} be such that
a(ni) − c(ni) are not divisible by 4 for suciently large i. Then
there exists a function p(n) satisfying (∗) such that G(Gdist
ni
, p)
doesn't follow extended zero-one law.
16/22

30. ƒpe™i—l sets of verti™es
he(nitionF †erti™es v1, . . . , vt of — gr—ph G = (V, E) —re s—id to
form — spe™i—l tEset if there doesn9t exist — vertex v ∈ V —dj—™ent
to —ll of the verti™es v1, . . . , vtF
17/22

31. ƒpe™i—l sets of verti™es
he(nitionF †erti™es v1, . . . , vt of — gr—ph G = (V, E) —re s—id to
form — spe™i—l tEset if there doesn9t exist — vertex v ∈ V —dj—™ent
to —ll of the verti™es v1, . . . , vtF
vet Rt ˜e — property of sp—nning su˜gr—phs of GnX
for —ny verti™es v1, . . . , vt not forming — spe™i—l tEset in Gn —nd
for —ny su˜set U ⊆ {v1, . . . , vt} there exists — vertex v
—dj—™ent to —ll verti™es from U —nd nonE—dj—™ent to —ll verti™es
from {v1, . . . , vt} UF
17/22

32. ƒpe™i—l sets of verti™es
he(nitionF †erti™es v1, . . . , vt of — gr—ph G = (V, E) —re s—id to
form — spe™i—l tEset if there doesn9t exist — vertex v ∈ V —dj—™ent
to —ll of the verti™es v1, . . . , vtF
vet Rt ˜e — property of sp—nning su˜gr—phs of GnX
for —ny verti™es v1, . . . , vt not forming — spe™i—l tEset in Gn —nd
for —ny su˜set U ⊆ {v1, . . . , vt} there exists — vertex v
—dj—™ent to —ll verti™es from U —nd nonE—dj—™ent to —ll verti™es
from {v1, . . . , vt} UF
€roposition
For every t ∈ N the random graph G(Gdist
n , p) satisfyes Rt a.a.s.
17/22

33. €roof of zeroEone kEl—wsX spe™i—l sets of verti™es without edges
ƒuppose
@IA Gn = (Vn, En) doesn9t h—ve spe™i—l (t − 1)Esets
@PA G(Gn, p) s—tisfyes Rt —F—FsF
18/22

34. €roof of zeroEone kEl—wsX spe™i—l sets of verti™es without edges
ƒuppose
@IA Gn = (Vn, En) doesn9t h—ve spe™i—l (t − 1)Esets
@PA G(Gn, p) s—tisfyes Rt —F—FsF
€roposition
Let a sequence Gn = (Vn, En) satisfy (1), (2) and the following
conditions:
Gn has special t-sets,
for every special t-set any two of its vertices are non-adjacent.
Then the random graph G(Gn, p) follows zero-one (t + 1)-law.
18/22

35. €roof of zeroEone kEl—wsX spe™i—l sets of verti™es with edges
€roposition
Suppose Gn = (Vn, En) satises (1), (2) and for any vertices
v1, . . . , vi where i t one of the following holds:
for any vertex vi+1 such that v1, . . . , vi+1 can be extended to
a special t-set there exist Ω(|Vn|β) dierent vertices each of
which can be mapped onto vi+1 by an automorphism of Gn
xing v1, . . . , vi (where β is a positive constant),
|{(vi+1, . . . , vt) : {v1, . . . , vt} is a special t-set}| = O(1).
Then the random graph G(Gn, p) follows extended zero-one
(t + 1)-law.
19/22

36. hisproof of extended zeroEone kEl—ws
vet L ˜e — property of su˜gr—phs G ⊆ GnX
for —ny (v1, . . . , vi) th—t ™—n ˜e extended to — spe™i—l tEset with
edges in Gn there exist vi+1, . . . , vt extending (v1, . . . , vi) to —
spe™i—l tEset with edges in GF
20/22

37. hisproof of extended zeroEone kEl—ws
vet L ˜e — property of su˜gr—phs G ⊆ GnX
for —ny (v1, . . . , vi) th—t ™—n ˜e extended to — spe™i—l tEset with
edges in Gn there exist vi+1, . . . , vt extending (v1, . . . , vi) to —
spe™i—l tEset with edges in GF
vet K(v1, . . . , vi) ˜e the num˜er of (vi+1, . . . , vt) extending
(v1, . . . , vi) to — spe™i—l tEset with edges in GnF
20/22

38. hisproof of extended zeroEone kEl—ws
vet L ˜e — property of su˜gr—phs G ⊆ GnX
for —ny (v1, . . . , vi) th—t ™—n ˜e extended to — spe™i—l tEset with
edges in Gn there exist vi+1, . . . , vt extending (v1, . . . , vi) to —
spe™i—l tEset with edges in GF
vet K(v1, . . . , vi) ˜e the num˜er of (vi+1, . . . , vt) extending
(v1, . . . , vi) to — spe™i—l tEset with edges in GnF
sf there exists (v1, . . . , vi) with
K(v1, . . . , vi) → ∞, K(v1, . . . , vi) = |Vn|o(1)
,
then PGn,p(L) ™—n —ppro—™h —ny num˜er from (0, 1)F
20/22

39. hisproof of extended zeroEone kEl—ws
‚epl—™e L ˜y — (rstEorder property LX
L = ∀v1 . . . ∀vi ∃vi+1 . . . ∃vt Q(v1, . . . , vt),
where Q —pproxim—tely s—ys th—t either (v1, . . . , vi) ™—n9t ˜e
extended to — spe™i—l tEset with edges in Gn or (v1, . . . , vt) forms —
spe™i—l tEset with edges in G(Gn, p)F
21/22

40. „h—nks
Thank you for your attention!
22/22