Bounded quantifier depth spectra for random graphs

Discrete Mathematics 339 (2016) 1651–1664

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Bounded quantifier depth spectra for random graphs J.H. Spencer a , M.E. Zhukovskii b,∗ a

Courant Institute, New York University, United States

b

Moscow Institute of Physics and Technology, Russian Federation

article

info

Article history: Received 13 January 2015 Received in revised form 1 August 2015 Accepted 8 January 2016 Available online 17 February 2016

abstract For which α there are first order graph statements A of given quantifier depth k such that a Zero–One law does not hold for the random graph G(n, p(n)) with p(n) at or near (there are two notions) n−α ? A fairly complete description is given in both the near dense (α near zero) and near linear (α near one) cases. © 2016 Elsevier B.V. All rights reserved.

Keywords: Random graphs Zero-one laws First-order logic Spectra

1. Introduction Asymptotic behavior of first-order properties probabilities of the Erdős–Rényi random graph G(n, p) have been widely studied in [7,3,12,9,8,13,16,15,17,18,27,11,21,20,22,24–26] (especially, the surveys [18,27] contain a description of all the main respective results). In [13] Shelah and the senior author showed that when α is an irrational number and p(n) = n−α+o(1) then G(n, p) obeys a Zero–One Law. (To avoid trivialities we shall restrict ourselves to 0 < α < 1.) In a series of papers [21,20,22,24–26] the junior author has examined when there is a Zero–One Law for all first order sentences of quantifier depth at most k. (In such cases we say that G(n, p) obeys Zero–One k-Law.) We here consider two notions of spectra, relative to k. We assume familiarity with the Erdős–Rényi random graph G(n, p) and of threshold functions (see [27,10,4,1]). We further assume familiarity with the first order language for graphs (see [18,27,2,5,19]). The quantifier depth of a sentence L is the number of nested quantifiers [27,19]. We let Lk denote the set of sentences L with quantifier depth at most k. As illustrative examples, the existence of a K4 has threshold function n−2/3 . The property that every pair x1 , x2 of vertices √ have a common neighbor y has threshold function n−1/2 ln n. For any first order property L we define two notions of its spectra, S 1 (L) and S 2 (L). The first considers behavior at p = n−α . 1 S (L) is the set of α ∈ (0, 1) which do not satisfy the following property: With p(n) = n−α , limn→∞ Pr[G(n, p(n)) |H L] exists and is either zero or one. The second considers behavior near p = n−α . S 2 (L) is the set of α ∈ (0, 1) which do not satisfy the following property: There exists ϵ > 0 so that for any n−α−ϵ < p(n) < n−α+ϵ , limn→∞ Pr[G(n, p(n)) |H L] = δ exists, is either zero or one, and is independent of the choice of p(n). Tautologically S 1 (L) ⊂ S 2 (L) but we need not have equality. Letting L be the sentence that every two vertices have a common neighbor, S 2 (L) = { 12 } while S 1 (L) = ∅.

∗

Corresponding author. E-mail address: [email protected] (M.E. Zhukovskii).

http://dx.doi.org/10.1016/j.disc.2016.01.005 0012-365X/© 2016 Elsevier B.V. All rights reserved.

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Definition 1. Let k ≥ 1. Sk1 is the union of all S 1 (L) where L ∈ Lk . Sk2 is the union of all S 2 (L) where L ∈ Lk . A full description of Sk1 and Sk2 appears difficult. Our main (though not only) concern shall be the values α of Sk1 and Sk2 that lie either near zero or near one. 2. Previous results Theorem 2 ([13]). Every S 2 (L) consists only of rational values α (as S 1 (L) ⊆ S 2 (L), the same is true for S 1 (L)). Moreover,  1 L∈L S (L) = Q ∩ (0, 1). In [21,20,22,24,25] some rational points from the set (0, 1) \ Sk1 were obtained. Theorem 3 ([21]). Let k ≥ 3 be an arbitrary natural number. If α ∈ (0, k−1 2 ) then the random graph G(n, n−α ) obeys Zero–One k-Law. Moreover, k−1 2 ∈ Sk1 . From this result it follows that the minimal number in Sk1 equals k−1 2 . We also obtain the maximal number in Sk1 . Theorem 4 ([22]). Let k > 3 be an arbitrary natural number. Let Q be the set of positive rational numbers with the numerator less than or equal to 2k−1 . The random graph G(n, n−α ) obeys the Zero–One k-Law, if α = 1 − 2k−11 +β , β ∈ (0, ∞)\ Q. Moreover,

for any β ∈ {1, . . . , 2k−1 − 2} 1−

1 2k−1 + β

∈ Sk1 .

Note that this result implies the following statement. For any k > 3, α > 1 − the Zero–One k-Law, if α ̸∈ {1 − known.

1 2k

,1 −

1 2 k −1

1 , 2 k −2

}. However, the maximal α such that G(n, n−α ) obeys the Zero–One k-Law is

Theorem 5 ([24]). Let k > 3 be an arbitrary natural number. Moreover, let α ∈ {1 − G(n, n

−α

the random graph G(n, n−α ) obeys

) obeys the Zero–One k-Law.

Hence the maximal number in Sk1 equals 1 −

1 2k

,1 −

1 2k −1

}. Then the random graph

1 . 2k −2

Recently, we extend the subset of the set Q from Theorem 4 such that for any β from this subset 1 − Theorem 6 ([26]). Let k > 4 be an arbitrary natural number. Moreover, let α = 1 − positive fraction with a ∈ {1, 2, . . . , 2 In [15] it was proved that sets

Sk1

k−1

− (b + 1) }. Then α ∈ 2

1 , 2k−1 +β

where β =

1 2k−1 +β a b

∈ Sk1 .

is an irreducible

Sk1 .

and Sk2 are infinite when k is large enough.

Theorem 7 ([15]). There exists k0 such that for any natural k > k0 sets Sk1 and Sk2 are infinite. There are, up to tautological equivalence, (see, e.g., [19]) only a finite number of first order sentences of a given quantifier j depth. Thus, for j either 1 or 2, set Sk is infinite if and only if there is a single L of quantifier depth at most k such that S j (L) is j

infinite. Therefore, we always search for one property with infinite spectrum when we prove that the spectrum Sk is infinite. It is also known [17] that all limit points of Sk1 and Sk2 are approached only from above. Theorem 8 ([17]). For any k ∈ N the set Sk2 is well-ordered under >. Consequently, the set Sk1 follows the same property. In this paper we try to answer the following questions. Q1 What are the maximal and the minimal numbers in Sk2 ? Q2 Let k be large enough so that sets Sk1 and Sk2 are infinite. What are the maximal and the minimal limit points in Sk1 and Sk2 ? 1 2 Q3 How many elements are there in  Sk and Sk near their minimal elements (the answer on this question for the maximal el-



j

ements is given in Theorem 6: Sk ∩ 1 −



1

2k−1

    , 1  = Ω (23k/2 ) for j ∈ {1, 2})? Consider, say, the interval I = 0, k−12.5 .

j

How many elements are there in Sk ∩ I, j ∈ {1, 2}? j Q4 For each j ∈ {1, 2} what is the minimal k such that Sk is infinite? 3. New results For any natural k we find the maximal and the minimal numbers in Sk2 and, therefore, answer the question Q1. Theorem 9. If k > 3, then min Sk2 = k−1 1 , max Sk2 = 1 −

1 . 2 k −2

Moreover, S32 = { 12 , 23 }.

J.H. Spencer, M.E. Zhukovskii / Discrete Mathematics 339 (2016) 1651–1664

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So, this result, Theorems 3–5 imply that for k > 3 the maximal numbers in Sk1 and Sk2 are equal, but the minimal numbers are not equal. We estimate the maximal and the minimal limit points in Sk1 and Sk2 and, therefore, partially answer the question Q2. j

j

For any j ∈ {1, 2} denote the set of limit points in Sk by (Sk )′ . Surprisingly, in both sets there are limit points near zero and near one. Theorem 10. There exists such k0 that for any natural k > k0 min(Sk1 )′ ≤

1 k − 11

,

min(Sk2 )′ ≤

As Sk2 ⊃ Sk1 , max(Sk2 )′ ≥ 1 − j



j

min(Sk )′ = min Sk



1 2k−13

j Sk ′

1 − log max( ) = (1 −

,

max(Sk1 )′ ≥ 1 −

1 2k−13

.

as well. Note that from Theorems 3–5, 9 and 10 it follows that for any j ∈ {1, 2}

  1+O

1 k−7

1 k

j log max Sk

, 

) 1+O

  1 k

.

Answering the question Q3 we estimate the number of elements in Sk1 and Sk2 in the interval 0, k−12.5 .





Theorem 11. For any j ∈ {1, 2}

     j 1  = Ω (k2k ). S ∩ 0,  k k − 2.5  Moreover, if k > 3 then Sk2 ∩ (0, k−1 2 ) = { k−1 1 , k−11.5 }. j

j





Note that we do not know if sets Sk ∩ 0, k−12.5 , Sk ∩ 1 − 2k1−1 , 1 are finite or not. Finally, we estimate the minimal k such that sets Sk1 and Sk2 are infinite.





Theorem 12. The minimal k1 and k2 such that Sk11 and Sk22 are infinite are from {4, . . . , 12} and {4, . . . , 10} respectively. This result is the first step in answering the question Q4. Remark. Our results concentrate at values of α either near zero or near one. It would be of interest to characterize α in other ranges, for example, α near 1/2. However, a full description of all α ∈ (0, 1) appears beyond our reach. Moreover, the j j j problems of finding the exact values of min(Sk )′ , max(Sk )′ , max{k : (Sk )′ = ∅} are still open. 4. Proofs Commonly, proofs of Zero–One Laws are based on searching of a winning strategy of the second player in Ehrenfeucht game, which is defined in the next section. 4.1. Ehrenfeucht game Let us define the game EHR(G, H , k) with two graphs G and H, two players (Spoiler and Duplicator) and a fixed number of rounds k. It is called the Ehrenfeucht game (see [10,1,19,6]). At the ν th round (1 ≤ ν ≤ k) Spoiler chooses either a vertex xν ∈ V (G) or a vertex yν ∈ V (H ). Duplicator chooses a vertex of the other graph. If Spoiler chooses at the µth round the vertex xµ ∈ V (G), xµ = xν (ν < µ), then Duplicator must choose yν ∈ V (H ). If at this round Spoiler chooses a vertex xµ ∈ V (G), xµ ̸∈ {x1 , . . . , xµ−1 }, then Duplicator must choose a vertex yµ ∈ V (H ), yµ ̸∈ {y1 , . . . , yµ−1 }. If Duplicator cannot do it, then Spoiler wins the game. At the end of the game the vertices x1 , . . . , xk ∈ V (G); y1 , . . . , yk ∈ V (H ) are chosen. Choose maximal collections of pairwise distinct vertices: xh1 , . . . , xhl ; yh1 , . . . , yhl , l ≤ k. Duplicator wins if and only if the mapping G|{xh ,...,xh } → H |{yh ,...,yh } which maps each xhν to yhν is an isomorphism. l l 1 1 The proofs of Zero–One Laws are based on the following statement (see [17,18,21,20,22,24,25,10,1]). Theorem 13. The equality lim Pr(Duplicator wins the game EHR(G(n, p(n)), G(m, p(m)), k)) = 1

n,m→∞

holds if and only if the random graph G(n, p) follows the Zero–One k-Law.

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J.H. Spencer, M.E. Zhukovskii / Discrete Mathematics 339 (2016) 1651–1664

We also use the following obvious extension of the theorem above in our proofs. Let p1 = p1 (n), p2 = p2 (n) be two functions with values in [0, 1] such that p1 (n) < p2 (n) for any n ∈ N. Theorem 14. If the equality lim Pr(Duplicator wins the game EHR(G(n, p(n)), G(m, p(m)), k)) = 1

n,m→∞

holds for any p ∈ (p1 , p2 ) then for any first order property L ∈ Lk there exists such δ ∈ {0, 1} that limn→∞ Pr(G(n, p) |H L) = δ for any p ∈ (p1 , p2 ). Moreover, if for any first order property L ∈ Lk there exists such δ ∈ {0, 1} that limn→∞ Pr(G(n, p) |H L) = δ for any p ∈ (p1 , p2 ) then a.a.s. Duplicator wins the game EHR(G(n, p(n)), G(m, p˜ (m)), k) for any p, p˜ ∈ (p1 , p2 ). 4.2. Small subgraphs Here we remind results about the distribution of the number of small subgraphs in G(n, p), which we exploit in our proofs. e(G) For an arbitrary graph G = (E , V ) denote e(G) = |E |, v(G) = |V |, ρ(G) = v(G) , ρ max (G) = maxH ⊆G ρ(H ). Denote the number of copies of G in G(n, p) by NG . Denote by LG the property of containing the graph G. Theorem 15 ([3,12]). If p = o n−1/ρ



max (G)



, then

lim Pr(G(n, p)  LG ) = 0.

n→∞

If n−1/ρ

max (G)

= o(p), then NG

lim Pr(G(n, p)  LG ) = 1,

En,p NG

n→∞

P

−→ 1.

Let G be a strictly balanced graph (its density ρ(G) is greater than a density of any its subgraph) with a(G) automorphisms. Theorem 16 ([3]). If p = n−1/ρ(G) then



d

NG −→ Pois



1 a(G)

.

4.3. Extensions To prove that the strategies of Duplicator which are described in the next section are winning it is required to state two theorems of the senior author about the number of extensions. Consider a graph Γ . Let (G, H ) be a pair of graphs G and H such that H ⊆ G ⊂ Γ , V (H ) = {x1 , . . . , xm }, V (G) = {x1 , . . . , xl } and set E (G) \ (E (H ) ∪ E (G \ H )) is non-empty. Denote e(G, H ) = e(G) − e(H ), v(G, H ) = v(G) − v(H ), e(G,H ) ρ(G, H ) = v( , ρ max (G, H ) = maxH ⊂K ⊆G ρ(K , H ). Moreover, let emax (G, H ) be the minimal number e(K , H ) over all K G ,H )

such that H ⊂ K ⊆ G, ρ(K , H ) = ρ max (G, H ) and set E (K ) \ (E (H ) ∪ E (K \ H )) is non-empty. Consider graphs  H, G such       that V (H ) = { x1 , . . . , xm }, V (G) = { x1 , . . . , xl }, H ⊂ G. The graph G is called a (G, H )-extension of graph H if

{xi1 , xi2 } ∈ E (G) \ E (H ) ⇒ { xi1 , xi2 } ∈ E ( G) \ E ( H ). We consider the property L(G,H ) of a graph to contain a (G, H )-extension of any its subgraph  H on m vertices. Theorem 17 ([16]). There exist 0 < ε < K such that if p ≤ ε n−1/ρ if p ≥ Kn

max (G,H )

−1/ρ max (G,H )

(ln n)1/e

max (G,H )

(ln n)

1/emax (G,H )

,

then lim Pr(G(n, p)  L(G,H ) ) = 0;

,

then lim Pr(G(n, p)  L(G,H ) ) = 1.

n→∞

n→∞

Consider vertices  x1 , . . . , xm ∈ Vn . Let N(G,H ) ( x1 , . . . , xm ) be a number of all (G, H )-extensions of G(n, p)|{x1 ,...,xm } in G(n, p). Theorem 18 ([14]). Let ε > 0. Then there exist t ∈ N and K > 0 such that if p ≥ Kn−1/ρ

max (G,H )

(ln n)1/t ,

then lim Pr(∀ x1 . . . ∀ xm (1 − ε)nv(G,H ) pe(G,H ) < N(G,H ) ( x1 , . . . , xm ) < (1 + ε)nv(G,H ) pe(G,H ) ) = 1.

n→∞

J.H. Spencer, M.E. Zhukovskii / Discrete Mathematics 339 (2016) 1651–1664

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Fig. 1. Maximal pair.

Further we use a corollary from Theorems 17 and 18 which exploits the following definitions. Let α ∈ (0, 1). Set fα (G, H ) = v(G, H ) − α e(G, H ). If the inequality fα (S , H ) > 0 holds for any graph S such that H ⊂ S ⊆ G, then the pair (G, H ) is called α -safe (see [10,1]). If the inequality fα (G, S ) < 0 holds for any graph S such that H ⊆ S ⊂ G, then the pair (G, H ) is called α -rigid (see [10,1]). For any natural number a we denote claα (G) (see [1]) a subgraph in Γ (it is called a-closure) such that there exist s ∈ Z+ and graphs G0 , G1 , . . . , Gs with the following property: G = G0 ⊂ G1 ⊂ · · · ⊂ Gs = claα (G), for any i ∈ {1, . . . , s} inequalities fα (Gi , Gi−1 ) < 0 and v(Gi , Gi−1 ) ≤ a hold and there is no subgraph  G in Γ such that fα ( G, Gs ) < 0 and v( G, Gs ) ≤ a. Before the corollary is formulated we recall the following lemma about closures from [1]. Lemma 1. Let p ≤ n−α , 0 < α < 1. Then for any natural numbers a and m there exists such D that a.a.s. for any  G ⊂ G(n, p) with v( G) = m inequality v(claα ( G)) < D holds. Introduce the definition of a maximal pair (see Fig. 1). Let T ⊂ K , |V (T )| ≤ |V (G)|. The pair (G, H ) is called (K , T )-maximal in Γ if for any subgraph  T of G such that |V ( T )| = |V (T )| and  T ∩ H ̸=  T the following property holds. There is no strict (K , T )-extension  K of  T in Γ \ (G \  T ) such that each vertex of V ( K ) \ V ( T ) is not adjacent to any vertex of V (G) \ V ( T ). The graph G is called (K , T )-maximal in Γ if for any subgraph  T of G with |V ( T )| = |V (T )| there is no strict (K , T )-extension  K of  T in Γ \ (G \  T ) such that each vertex of V ( K ) \ V ( T ) is not adjacent to any vertex of V (G) \ V ( T ). Corollary 1. Let 0 < α1 < α2 < 1. Let a pair (G, H ) be α2 -safe and K be a finite set of pairs of graphs such that for any (K , T ) ∈ K inequalities fα1 (K , T ) < 0 and v(T ) ≤ v(G) hold. Let p ∈ [n−α2 , n−α1 ]. Then a.a.s. for any  x1 , . . . , xm in G(n, p)

˜ of H˜ := G(n, p)|{x1 ,...,xm } such that for any (K , T ) ∈ K the pair ( there is a strict (G, H )-extension G G,  H ) is (K , T )-maximal.

Proof. Let  x1 , . . . , xm ∈ Vn . Let N(KG,H ) ( x1 , . . . , xm ) be a number of all (G, H )-extensions  G of  H := G(n, p)|{x1 ,...,xm } in G(n, p) (K ,T )

such that for any (K , T ) ∈ K the pair ( G,  H ) is (K , T )-maximal in G(n, p). For any (K , T ) ∈ K denote by N(G,H ) ( x1 , . . . , xm )

a number of all (G, H )-extensions  G of  H := G(n, p)|{x1 ,...,xm } in G(n, p) such that the pair ( G,  H ) is not (K , T )-maximal in G(n, p). Fix (K , T ) ∈ K . Let there be graphs R = R(K , T ), S1 , S2 such that H ⊂ R ⊆ G, S1 ⊆ T , S1 ⊂ S2 ⊆ (K \ (T \ S1 )),  the set of all such pairs (K , T ) and by R(K , T ) the set of v(R) ≥ v(S1 ) and fα1 (R, H ) + fα1 (S2 , S1 ) < 0 (we denote by K graphs R(K , T ) which follow the above properties). As p ≤ n−α1 , then by Lemma 1 there exists such D(R, K , T ) that a.a.s. (K ,T )

∀ x1 . . . ∀ xm

N(R,H ) ( x1 , . . . , xm ) < D(R, K , T ).

Let  R be any graph such that H ⊂  R ⊆ G. Then inequality fα2 ( R, H ) > 0 holds. Therefore, for any M  p ≥ n−α2 ≫ n−1/ρ(R,H ) (ln n)M .

Consequently, for any K and any M inequality p ≥ Kn−1/ρ (G,H ) (ln n)M holds for n large enough. It is easy to conclude from Theorem 18 and Corollary 1 that for any N and any ε > 0 a.a.s. for any vertices  x1 , . . . , xm , xm+1 , . . . , xm+N there exist at least (1 − ε)nv(G,H ) pe(G,H ) strict (G, H )-extensions of G(n, p)|{x1 ,...,xm } in G(n, p)|Vn \{xm+1 ,...,xm+N } . Therefore, setting max

N =





D(R, K , T )v(R, H )

 R∈R(K ,T ) (K ,T )∈K

we obtain that in G(n, p) for any ε > 0 a.a.s.

∀ x1 . . . ∀ xm N(KG,H ) ( x1 . . . xm ) > (1 − ε)nv(G,H ) pe(G,H ) . 

(1)

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. Let there be graphs R = R(K , T ), S1 , S2 and α ∈ [α1 , α2 ] such that H ⊂ R ⊆ G, S1 ⊆ T , Let (K , T ) ∈ K \ K  the set of all such pairs (K , T )). As for S1 ⊂ S2 ⊆ (K \ (T \ S1 )), v(R) ≥ v(S1 ) and fα (R, H ) + fα (S2 , S1 ) = 0 (we denote by K any pair of graphs (A, B) and any number D the property to contain for any subgraph  B of G(n, p) at most D (A, B)-extensions of  B is monotone, Theorem 18 and Lemma 1.10 from [10] imply that for any δ > 0 small enough and any p ∈ [n−α−δ , n−α+δ ] a.a.s. (K ,T ) ∀ x1 . . . ∀ xm N(G,H ) ( x1 , . . . , xm ) < 2

v(G)! nv(G,H )+v(K ,T ) n(−α+δ)(e(G,H )+e(K ,T )) (v(G) − v(T ))!  −α e(G,H )

= nv(G,H ) pe(G,H ) nfα (K ,T )

n

nδ(e(G,H )+e(K ,T ))

p

≤ nv(G,H ) pe(G,H ) nfα (K ,T ) nδ(2e(G,H )+e(K ,T )) .

(2)

(G, H ) ∪ K (G, H )). Then for any S1 , S2 , S1 ⊆ T , S1 ⊂ S2 ⊆ (K \ (T \ S1 )), any R, H ⊂ R ⊆ G, such that Let (K , T ) ∈ K \ (K v(R) ≥ v(S1 ) and any M v(R,H )+v(S2 ,S1 ) − p ≥ n−α2 ≫ n e(R,H )+e(S2 ,S1 ) (ln n)M

as for any (A, B) the function fα (A, B) is continuous in α on (α1 , α2 ). Therefore, by Theorem 18 a.a.s.

v(G)! nv(G,H )+v(K ,T ) pe(G,H )+e(K ,T ) (v(G) − v(T ))!   = O nv(G,H ) pe(G,H ) nfα1 (K ,T ) .

(K ,T ) ∀ x1 . . . ∀ xm N(G,H ) ( x1 , . . . , xm ) < 2

Inequalities (1), (2), fα1 (K , T ) < 0 and finiteness of the set K imply



lim Pr ∀ x1 . . . ∀ xm N(KG,H ) ( x1 , . . . , xm ) > n→∞

1 v(G,H ) e(G,H ) n p 2



= 1.

Corollary 1 is proved.

4.4. Proof of Theorem 9 First of all, let us prove that for any k ≥ 3 the numbers k−1 1 , k−11.5 are in Sk2 . Consider the first-order property L ∈ Lk expressed by the formula

∀x1 · · · ∀xk−1 ∃xk (xk ∼ x1 ) ∧ · · · ∧ (xk ∼ xk−1 ). Let ε be any number from (0, k−1 1 ). From Theorem 17 it follows that lim Pr(G(n, n−1/(k−1)+ε )  L) = 1,

n→∞

lim Pr(G(n, n−1/(k−1)−ε )  L) = 0.

n→∞

Therefore, k−1 1 ∈ Sk2 . Further, consider the first-order property L ∈ Lk expressed by the formula

∀x1 . . . ∀xk−2 ∃xk−1 ∃xk (xk−1 ∼ x1 ) ∧ · · · ∧ (xk−1 ∼ xk−2 ) ∧ (xk ∼ x1 ) ∧ · · · ∧ (xk ∼ xk−1 ). Let ε be any number from (0, k−11.5 ). From Theorem 17 it follows that lim Pr(G(n, n−1/(k−1.5)+ε )  L) = 1,

n→∞

lim Pr(G(n, n−1/(k−1.5)−ε )  L) = 0.

n→∞

Therefore, k−11.5 ∈ Sk2 as well. Secondly, we prove that (0, k−1 1 ) ∪ ( k−1 1 , k−11.5 ) ∪ ( k−11.5 , k−1 2 ) ∩ Sk2 = ∅ for any k ≥ 3. This implies that min Sk2 = k−1 1 , 2 S3 = { 13 , 23 }. Moreover, this implies the second statement of Theorem 11. We start from the interval (0, k−1 1 ). Let α ∈ (0, k−1 1 ), ε ∈ (0, min{α, k−1 1 − α}). It was proved in [21] (proof of Theorem 3) that for any p ∈ (n−α−ε , n−α+ε ) a.a.s. Duplicator has a winning strategy in EHR(G(n, p(n)), G(m, p(m)), k). Therefore, Theorem 14 implies α ̸∈ Sk2 .

J.H. Spencer, M.E. Zhukovskii / Discrete Mathematics 339 (2016) 1651–1664

1657

Let α1 , α2 ∈ ( k−1 1 , k−11.5 ), α1 < α2 and p ∈ [n−α2 , n−α1 ]. Moreover, let SA be a set of all graphs Γ that satisfy the following properties. A1 The full level k − 2 extension property (see [1]): for any non-negative integer numbers a, b such that a + b ≤ k − 2 and any vertices v1 , . . . , va , u1 , . . . , ub of Γ there exists a vertex x ∈ V (Γ ) such that x ∼ v1 , . . . , x ∼ va , x  u1 , . . . , x  ub . ˜ ⊂ Γ such that v(H˜ ) = v(H ) there is a strict A2 For any pair of graphs (G, H ) with v(H ) = k − 2, v(G, H ) ≤ 2 and any H (G, H )-extension of H˜ in Γ . A3 Let (K , T ) be a α1 -rigid pair such that v(T ) = k − 1, v(K , T ) = 1. Then for any pair (G, H ) with v(H ) = k − 2, v(G, H ) = 1 ˜ ⊂ Γ with v(H˜ ) = v(H ) there is a strict (G, H )-extension G˜ of H˜ in Γ such that the pair (G˜ , H˜ ) is and any graph H (K , T )-maximal. Let A, B ∈ SA . In [21] it was proved that Duplicator has a winning strategy in EHR(A, B, k). Moreover, it was proved in [21] that a.a.s. G(n, p) satisfies A1. It remains to apply Corollary 1 for the properties A2 and A3. Indeed, any pair of graphs (G, H ) with v(H ) = k − 2, v(G, H ) ≤ 2 is α2 -safe. Applying Corollary 1 for any such pair (G, H ) we get the properties A2 and A3 a.a.s. Therefore, lim Pr(G(n, p) ∈ SA ) = 1.

n→∞

By Theorem 14 any α ∈ ( k−1 1 , k−11.5 ) is not in Sk2 . Finally, let α1 , α2 ∈ ( k−11.5 , k−1 2 ), α1 < α2 , p ∈ [n−α2 , n−α1 ] and SB be a set of all graphs Γ that satisfy the following properties. B1 The full level k − 3 extension property. B2 Let K = {(K1 , T1 ), (K2 , T2 )} be a set of α1 -rigid pairs, where v(T1 ) = k−2, v(K1 , T1 ) = 2, v(T2 ) = k−1, v(K2 , T2 ) = 1. For ˜ ⊂ Γ such that v(H˜ ) = v(H ) there is a strict (G, H )-extension any α2 -safe pair of graphs (G, H ) with v(G) ≤ k and any H ˜ of H in Γ . Let A, B ∈ SB . As in the previous case, applying statements from the proof of Theorem 3 from [21], Corollary 1 and Theorem 14 it is easy to prove that limn→∞ Pr(G(n, p) ∈ SB ) = 1 and, consequently, α ∈ ( k−11.5 , k−1 2 ) is not in Sk2 . We finish the proof with considering α ∈ (1 −



1−

1 2k −1

,1 −

1 2k





∪ 1−

1 2k

 , 1 ) . In [23] it was proved that if α ∈ I ( k ) := 1− 2k −2 1

,1 − 2 k −2 1



1 2k −1

∪



, 1 , then there are numbers n1 = n1 (α, k), n2 = n2 (α, k), n3 = n3 (k) and n4 such

that the following property holds. If graphs A, B are (n1 , n2 , n3 , n4 , 1/α)-sparse, then Duplicator has a winning strategy in EHR(A, B, k) (this strategy is also described in [24], see Section 4.8). We call a graph Γ (n1 , n2 , n3 , n4 , 1/α)-sparse, if it satisfies two properties. C1 For any graph K with v(K ) ≤ n1 and ρ max (K ) < 1/α in Γ there exists a subgraph isomorphic to K , for any graph K with v(K ) ≤ n1 and ρ max (K ) > 1/α in Γ there are no subgraphs isomorphic to K . C2 Let H be a set of α -safe pairs (H1 , H2 ) such that v(H1 ) ≤ n1 , v(H2 ) ≤ n2 . Let K be a set of α -rigid pairs (K1 , K2 ) such that v(K1 ) ≤ n3 , v(K2 ) ≤ n4 . For any pairs (H1 , H2 ) ∈ H , (K1 , K2 ) ∈ K and any graph G2 ⊂ Γ such that v(G2 ) = v(H2 ) in Γ there is a strict (H1 , H2 )-extension G1 of G2 such that the pair (G1 , G2 ) is (K1 , K2 )-maximal in Γ . Let us prove that for any α ∈ I (k) there exists ε > 0 such that (α − ε, α + ε) ∈ I (k) and for any p ∈ (n−α−ε , n−α+ε ) a.a.s. G(n, p) is (n1 , n2 , n3 , n4 , 1/α)-sparse. Let us start with the property C1. Consider a set G of pairwise non-isomorphic graphs with at most n1 vertices and a maximal density which is not equal to 1/α such that any graph G with v(G) ≤ n1 and ρ max (G) ̸= 1/α is isomorphic to some graph from G. Let G1 be a set of pairwise non-isomorphic graphs with at most n1 vertices and maximal density less than 1/α such that any graph which satisfies these properties is isomorphic to some graph from G1 . Obviously, |G1 | ≤ |G| < ∞. Let α − ε > maxG∈G\G1 1/ρ max (G), α + ε < minG∈G1 1/ρ max (G). Then by Theorem 15 for any p ∈ (n−α−ε , n−α+ε ) equalities lim Pr(∀G ∈ G1 G(n, p)  LG ) = 1,

n→∞

lim Pr(∃G ∈ G \ G1 G(n, p)  LG ) = 0

n→∞

hold. The property C1 is proved. Let H be a set of pairwise non-isomorphic α -safe pairs (H1 , H2 ) such that v(H1 ) ≤ n1 , v(H2 ) ≤ n2 and the cardinality of H is maximal (we call two pairs of graphs (A1 , B1 ) and (A2 , B2 ) such that B1 ⊂ A1 , B2 ⊂ A2 isomorphic if the graphs (V (A1 ), E (A1 ) \ E (B1 )) and (V (A2 ), E (A2 ) \ E (B2 )) are isomorphic). Let K be a set of pairwise non-isomorphic α -rigid pairs (K1 , K2 ) such that v(K1 ) ≤ n3 , v(K2 ) ≤ n4 and the cardinality of K is maximal. Let α − ε > max(K1 ,K2 )∈K ρ(K 1,K ) ,

α + ε < min(H1 ,H2 )∈H

1

ρ max (H1 ,H2 )

. Then by Corollary 1 for any p ∈ (n−α−ε , n−α+ε ) a.a.s. for any (H1 , H2 ) ∈ H

1

2

∀ x1 ...∀ xv(H2 ) N(KH1 ,H2 ) ( x1 , . . . , xv(H2 ) ) > 0. Therefore, the property C2 is proved. Thus, by Theorem 14 the set I (k) ∩ Sk2 is empty. Finally, let α ∈ {1 − 2k1−1 , 1 − 21k }. Let us define a set of graphs SD . A graph Γ is in SD if and only if the following properties hold.

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J.H. Spencer, M.E. Zhukovskii / Discrete Mathematics 339 (2016) 1651–1664

D1 In Γ there are no strictly balanced subgraphs (its density is greater than a density of any its proper subgraph) with at most 2k+1 vertices and density greater than 1/α . D2 Let H be a set of all α -safe pairs (H1 , H2 ) (up to isomorphism) such that v(H1 ) ≤ k2k . Then for any (H1 , H2 ) ∈ H and any G ⊂ Γ , v(G) = v(H2 ), there is a strict (H1 , H2 )-extension of G in Γ . D3 Let H be a (finite) set of α -rigid pairs (H1 , H2 ) such that v(H1 ) ≤ 2k , v(H2 ) ≤ 2. Then for any strictly balanced graph H with ρ(H ) < 1/α with at most 2k vertices in Γ there is a copy G of H such that there is no (H1 , H2 )-extension of any subgraph G1 of G in Γ \ (G \ G1 ) for any (H1 , H2 ) ∈ H . D4 Let H be a (finite) set of α -safe pairs (H1 , H2 ) such that v(H1 ) ≤ k2k . Let K be a (finite) set of α -rigid pairs (K1 , K2 ) such that v(K1 ) ≤ 2k , v(K2 ) ≤ 2. Then for any pair (H1 , H2 ) ∈ H and any subgraph G2 ⊂ Γ with v(H2 ) vertices in Γ there is a strict (H1 , H2 )-extension G1 of G2 such that (G1 , G2 ) is (K1 , K2 )-maximal in Γ for any (K1 , K2 ) ∈ K . In [23] it was proved that if A, B ∈ SD , then Duplicator has a winning strategy in EHR(A, B, k). Let us prove that there exists ε > 0 such that for any p ∈ (n−α−ε , n−α+ε ) a.a.s. G(n, p) ∈ SD . It can be shown similarly to the case α ∈ I (k). The only difference is in the proof of property D3. It uses the idea from the proof of Corollary 1. We do not give its proof, we just give the idea: from Theorem 15 it follows that the number of ‘‘extended’’ by a rigid extension graphs is asymptotically smaller than the number of not ‘‘extended’’ graphs. By Theorem 14 the numbers 1 − 2k1−1 , 1 − 21k are not in Sk2 . Theorem 9 is proved. 4.5. Proof of Theorem 10 We start from the inequality min(Sk1 )′ ≤ k−111 . Let k ≥ 15. Let j ∈ N, m = j(k − 10), s= and p = n

k − 11 k − 10 −α

·

4(mm+1 − m) m−1

,

α=

1 k − 11

+

1 s

. Given vertices x1 , . . . , xk−11 , z, consider the following predicates:

Tz (x) = [∃v ((v ∈ N (x1 , . . . , xk−13 , z , x)) ∧ (∀y ∈ X ((y ̸= x) ⇒ (y  v))))], Rz (a, b) = [∃v ((v ∈ N (x1 , . . . , xk−14 , z , a, b)) ∧ (∀y ∈ X (((y ̸= a) ∧ (y ̸= b)) ⇒ (y  v))))], where X = N (x1 , . . . , xk−11 ) is the set of all common neighbors of x1 , . . . , xk−11 . Obviously, both predicates are expressed in the first-order language. Let Nz (x) be the set of neighbors of a vertex x under the adjacency Rz : Nz (x) = {a : Rz (a, x)}; H1 (z ) be the set of all vertices which satisfy Tz ; H2 (z ) be the set of all neighbors of vertices from H1 (z ) under the adjacency Rz . Let L(x1 , . . . , xk−11 ) be the property of vertices x1 , . . . , xk−11 , which is defined by the first-order formula of quantifier depth 11

φ = ∃z (ϕ1 (z ) ∧ ϕ2 (z ) ∧ ϕ3 (z ) ∧ ϕ4 (z )), where

ϕ1 (z ) = [∀u1 ∈ X ∀u2 ∈ X (((u1 ∈ H1 (z )) ∧ (u2 ∈ H1 (z )) ∧ (u1 ̸= u2 )) ⇒ ((Nz (u1 ) ̸=∗ Nz (u2 )) ∧ (Nz (u1 ) ∩ Nz (u2 ) = ∅) ∧ (Nz (u1 ) ∩ H1 (z ) = ∅)))], ϕ2 (z ) = [∀x ∈ X ((x ∈ H1 (z )) ∨ (x ∈ H2 (z )))], ϕ3 (z ) = [∀x ∈ X [((x ∈ H1 (z )) ∧ (∀y ∈ X ((y ∈ H1 (z )) ⇒ (Nz (x) ≤∗ Nz (y))))) ⇒ (H1 (z ) =∗ Nz (x))]], ϕ4 (z ) = [∀x ∈ X ∀y ∈ X ∀˜x ∈ X [((x ∈ H1 (z )) ∧ (y ∈ H1 (z )) ∧ (Nz (x) <∗ Nz (y)) ∧ (∀˜y ∈ X (((˜y ∈ H1 (z )) ∧ (Nz (x) <∗ Nz (˜y)) ∧ (y ̸= y˜ )) ⇒ (Nz (y) <∗ Nz (˜y)))) z

z

∧ (˜x ∈ H1 (z )) ∧ (∀˜y ∈ X ((˜y ∈ H1 (z )) ⇒ (Nz (˜x) ≤∗ Nz (˜y))))) ⇒ (Nz (y) = Nz (x) · Nz (˜x))]]. Here

(Nz (u1 ) ∩ Nz (u2 ) = ∅) = (∀y1 ∀y2 ((Rz (u1 , y1 ) ∧ Rz (u2 , y2 )) ⇒ (y1 ̸= y2 ))), (Nz (u1 ) ∩ H1 (z ) = ∅) = (∀y(Rz (u1 , y) ⇒ (¬Tz (y)))), (x ∈ H1 (z )) = Tz (x), (x ∈ H2 (z )) = (∃y(Rz (x, y) ∧ Tz (y))). The formula (A ≤∗ B) expresses the property that implies the inequality |A| ≤ |B|:

(A ≤∗ B) = (∃˜z (∀x[(x ∈ A \ B) ⇒ (∃y [(y ∈ B \ A) ∧ Rz˜ (x, y)])] ∧ [(x ∈ B \ A) ⇒ (¬(∃y∃˜y (y ̸= y˜ ) ∧ Rz˜ (x, y) ∧ Rz˜ (x, y˜ )))])). Moreover, we exploit the notations

(A =∗ B) = (A ≤∗ B) ∧ (B ≤∗ A), (A ≥∗ B) = (B ≤∗ A).

(A <∗ B) = ((A ≤∗ B) ∧ (¬(A =∗ B))),

(3)

J.H. Spencer, M.E. Zhukovskii / Discrete Mathematics 339 (2016) 1651–1664 z

1659

z

Finally, for any sets A, B, C the formula (A = B · C ) expresses the property that implies the equality |A| = |B||C |: z z (A = B · C ) = ([∀a ∈ A ∃b ∈ B ∃c ∈ C (Rz (a, b) ∧ Rz (a, c ) ∧ (∀b˜ ∈ B((b ̸= b˜ ) ⇒ (¬Rz (a, b˜ )))) ∧ (∀˜c ∈ C ((c ̸= c˜ ) ⇒ (¬Rz (a, c˜ )))))] ∧ [∀b ∈ B ∀c ∈ C ∃a ∈ A(Rz (a, b) ∧ Rz (a, c ) ∧ (∀˜a ∈ A ((˜a ̸= a) ⇒ ((¬Rz (˜a, b)) ∨ (¬Rz (˜a, c ))))))]).

Formula ϕ1 represents the property of sets Nz (u1 ), Nz (u2 ) and H1 (z ) to be pairwise disjoint for all distinct u1 , u2 ∈ H1 (z ) ∩ X . It also represents the property of sets Nz (u1 ), Nz (u2 ) to have different cardinalities. We order the set H1 (z ) according to this cardinality: (u1 < u2 ) ⇔ (|Nz (u1 )| < |Nz (u2 )|). Formula ϕ2 represents the property of any vertex of X to be either in H1 (z ) or in H2 (z ). Formula ϕ3 represents the property of the set Nz (u), which corresponds to the smallest element u in H1 (z ), to have cardinality |H1 (z )|. Formula ϕ4 represents the following property: if y is the successor of x in the described ordering and x˜ is the smallest element then |Nz (y)| = |Nz (x)||Nz (˜x)|. Obviously, the considered formula φ is of the first order. Consider the property L˜ (x1 , . . . , xk−11 ), which is expressed by the first-order formula with quantifier depth 7

∀y ((N (x1 , . . . , xk−11 ) ≥∗ N (x1 , . . . , xk−12 , y))). We are going to prove that the first order property there exist x1 , . . . , xk−11 such that L(x1 , . . . , xk−11 ) ∧ L˜ (x1 , . . . , xk−11 ) (which is expressed by the first order formula with quantifier depth k) holds with limiting probability from (0, 1). Let us say that the vertices x1 , . . . , xk−11 are moderate, if

|N (x1 , . . . , xk−11 )| = max |N (x1 , . . . , xk−12 , y)|. y

˜ n ⊂ Ωn contain all graphs which satisfy the following properties. Let Ω A1 For any r ≤ k−s11 , any x1 , . . . , xk−14 , any pairs (a1 , b1 ), . . . , (ar , br ), which contain distinct vertices, and any set of vertices Γ , which contains the vertices a1 , b1 , . . . , ar , br and does not contain any of x1 , . . . , xk−14 , there exists a vertex z such that for any i ∈ {1, . . . , r } predicate Rz (ai , bi ) is true and for any other pair of vertices (a, b) from Γ predicate Rz (a, b) is not true. A2 For any x1 , . . . , xk−12 there is a vertex xk−11 such that |N (x1 , . . . , xk−11 )| ≥ k−s11 . A3 For any x1 , . . . , xk−11

|N (x1 , . . . , xk−11 )| ≤ s. A4 There are no subgraphs with at most mm+1 vertices and density which is greater than 1/α .

˜ n ) = 1. From Theorem 15, Theorem 17 and Corollary 1 it follows that limn→∞ Pr(G(n, p) ∈ Ω ˜ n . From the properties A2, A3 it follows that for any moderate x1 , . . . , xk−11 Let G ∈ Ω |N (x1 , . . . , xk−11 )| ∈



s k − 11

 ,s .

(4)

˜ Then x1 , . . . , xk−11 are moderate and Let x1 , . . . , xk−11 satisfy L ∧ L. |N (x1 , . . . , xk−11 )| =

t t +1 − t t −1

+t

for some t ∈ N. If t < m then from (4) it follows that

(m − 1)m − m + 1 + m − 1 ≥ |N (x1 , . . . , xk−11 )| m−2 4 mm+1 − m m (m − 1)m − m + 1 ≥ · ≥ · . k − 10 m−1 k − 10 m−2 Therefore,

(m − 1)m − m + 1 (k − 10)(m − 1) ≤ . m−2 m − k + 10 We arrive at contradiction with equality m = j(k − 10). If t > m then from (4) it follows that (m + 1)m+2 − m − 1 m

+ m + 1 ≤ |N (x1 , . . . , xk−11 )| ≤4

k − 10 k − 11

·

mm+1 − m m−1

≤

4 m

·

(m + 1)m+2 − m − 1 m

.

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J.H. Spencer, M.E. Zhukovskii / Discrete Mathematics 339 (2016) 1651–1664 mm+1 −m m−1

We arrive at contradiction again. Thus, t = m. Note that

edges Rz in X are necessary for constructing the set H2 (z ),

m+1 2 2 m m−−1m

z

z

edges Rz in X are necessary for representing all predicates A = B · C , m predicates Tz are necessary for constructing the set H1 (z ). Therefore, in G there is the graph K (x1 , . . . , xk−11 ), with sets of vertices and edges which contain all the vertices and edges representing the described predicates. This graph consists of k − 11 +

mm+1 − m m−1

+m+m+1+

mm+1 − m m−1

+2

mm+1 − m2 m−1

vertices and

(k − 11)



mm+1 − m m−1

+m+m+

mm+1 − m m−1

+2

mm+1 − m2



m−1

edges. Obviously, it is strictly balanced and its density equals α1 . From Theorem 16 it follows that this subgraph is in G(n, p) with limiting probability c for some c ∈ (0, 1). Consequently, from Theorem 16 it follows that for any ε > 0 and n large enough Pr(G(n, p)  (∃x1 . . . ∃xk−11 (L(x1 , . . . , xk−11 ) ∧ L˜ (x1 , . . . , xk−11 )))) ≤ (1 + ε)c . Finally, if G contains K (x1 , . . . , xk−11 ) for some x1 , . . . , xk−11 then from A4 it follows that maxy̸=xk−11 |N (x1 , . . . , xk−12 , y)| ≤ s and, therefore, k−11

∀y ((|N (x1 , . . . , xk−11 )| ≥ |N (x1 , . . . , xk−12 , y|))). Moreover, A1 implies L˜ (x1 , . . . , xk−11 ) because we need maxy̸=xk−11 |N (x1 , . . . , xk−11 , y)| ≤

property A1 implies L(x1 , . . . , xk−11 ) as well because we need at most m Therefore, for any ε > 0 and n large enough

m−1

<

s k−11

s k−11

edges Rz˜ for this. The

edges Rz˜ for represent predicates ≤∗ .

Pr(G(n, p)  (∃x1 . . . ∃xk−11 (L(x1 , . . . , xk−11 ) ∧ L˜ (x1 , . . . , xk−11 )))) ≥ (1 − ε)c . Finally, lim Pr(G(n, p)  (∃x1 . . . ∃xk−11 (L(x1 , . . . , xk−11 ) ∧ L˜ (x1 , . . . , xk−11 )))) = c .

n→∞

The first inequality is proved. Let us prove that min(Sk2 )′ ≤

1 k−7

.

(5)

Let k ≥ 10. Let m ∈ N, s = m(k − 7) and k−1 7 + s+1 1 < α < k−1 7 + 1s , p = n−α . Given vertices x1 , . . . , xk−9 , z, consider the following predicate Rz (a, b) = (∃v ((v ∼ x1 ) ∧ · · · ∧ (v ∼ xk−10 ) ∧ (v ∼ z ) ∧ (v ∼ a) ∧ (v ∼ b)))

(6)

and the property ‘‘x1 , . . . , xk−9 are even-extendible’’, which is expressed by the sentence min max |N (x1 , . . . , xk−7 )| is even. xk−8 xk−7

˜ n ⊂ Ωn contain all graphs which satisfy the following properties. Let Ω B1 For any x1 , . . . , xk−10 , any natural l ≤ m, any pairs (a1 , b1 ), . . . , (al , bl ), which contain distinct vertices, and any set of vertices Γ , which contains vertices a1 , b1 , . . . , al , bl but does not contain any of x1 , . . . , xk−10 , there exists a vertex z such that for any t ∈ {1, . . . , l} predicate Rz (at , bt ) is true and for any other pair of vertices (a, b) from Γ predicate Rz (a, b) is not true. B2 For any x1 , . . . , xk−8 there is a vertex xk−7 such that |N (x1 , . . . , xk−7 )| ≥ m. B3 For any x1 , . . . , xk−9 there is a vertex xk−8 such that max |N (x1 , . . . , xk−9 , xk−8 , xk−7 )| = m. xk−7

˜ n ) = 1. We easily finish the prove of (5) after From Theorem 17 and Corollary 1 it follows that limn→∞ Pr(G(n, p) ∈ Ω the following statement. Lemma 2. There is a first order property L which is expressed by formula ϕ with quantifier depth k such that lim (Pr(G(n, p)  L) − Pr(any x1 , . . . , xk−9 of G(n, p) are even-extendible)) = 0.

n→∞

J.H. Spencer, M.E. Zhukovskii / Discrete Mathematics 339 (2016) 1651–1664

1661

Proof. Consider the property L which is expressed by the formula ϕ =

(∀x1 . . . ∀xk−9 ∃xk−8 ∃xk−7 (([∀v (N (x1 , . . . , xk−7 ) ≥∗ N (x1 , . . . , xk−8 , v))] ∧ [∀˜xk−8 ∃˜xk−7 (N (x1 , . . . , xk−7 ) ≤∗ N (x1 , . . . , xk−9 , x˜ k−8 , x˜ k−7 ))]) ⇒ (N (x1 , . . . , xk−7 ) is even))), where ≤∗ is defined in (3) and Rz is defined in (6),

(N (x1 , . . . , xk−7 ) is even) = (∃z ∀x ∈ N (x1 , . . . , xk−7 ) ∃y ∈ N (x1 , . . . , xk−7 ) ((x ̸= y) ∧ [Rz ,x1 ,...,xk−10 (x, y) ∧ (∀˜y ∈ N (x1 , . . . , xk−7 ) ((y ̸= y˜ ) ⇒ (¬Rz ,x1 ,...,xk−10 (x, y˜ ))))])).

(7)

˜ n and any its vertices x1 , . . . , xk−9 be even-extendible. Let x1 , . . . , xk−9 be arbitrary vertices of G. Consider a vertex Let G ∈ Ω xk−8 such that there exists a vertex xk−7 with the following property [∀v (|N (x1 , . . . , xk−7 )| ≥ |N (x1 , . . . , xk−8 , v|))] ∧ [∀˜xk−8 ∃˜xk−7 (|N (x1 , . . . , xk−7 )| ≤ |N (x1 , . . . , xk−9 , x˜ k−8 , x˜ k−7 )|)]. ˜ n it follows that we can replace the last predicate with its Then |N (x1 , . . . , xk−7 )| = m. Therefore, from the definition of Ω first-order analogue [∀v (N (x1 , . . . , xk−7 ) ≥ N (x1 , . . . , xk−8 , v))] ∧ [∀˜xk−8 ∃˜xk−7 (N (x1 , . . . , xk−7 ) ≤ N (x1 , . . . , xk−9 , x˜ k−8 , x˜ k−7 ))]. Moreover, the predicate ‘‘|N (x1 , . . . , xk−7 )| is even’’ can be replaced with ‘‘N (x1 , . . . , xk−7 ) is even’’ by the same reason. Therefore, L holds. Obviously, for any graph G ∈ Ωn which satisfies L any its vertices x1 , . . . , xk−9 are even-extendible. As ˜ n ) = 1, Lemma 2 is proved. limn→∞ Pr(G(n, p) ∈ Ω

˜ n then any vertices x1 , . . . , xk−9 of G are even-extendible. Therefore, limn→∞ Pr(G(n, p)  L) = 1. If m is even and G ∈ Ω ˜ n then any vertices x1 , . . . , xk−9 of G are not even-extendible. Therefore, limn→∞ Pr(G(n, p)  L) = 0. If m is odd and G ∈ Ω The inequality (5) is proved. Let us prove that max(Sk1 )′ ≥ 1 −

1 2k−13

.

(8)

Let k ≥ 16. Let m ∈ N and α = 1 − 2k−1 13 + 2k−114 m , p = n−α . Consider integers n1 ≤ n2 ≤ n3 such that n1 + n2 + n3 = 2k−12 and n3 − n1 = 1. Given a vertex z, consider the following predicate Rz (a, b) = (∃v (D∗n1 (a, v) ∧ D∗n2 (b, v) ∧ D∗n3 (z , v))),

(9)

where D∗i (x, y) = Di (x, y) ∧ ¬ ∨j
 



and Di (x, y) is the first order formula with quantifier depth ⌈log2 i⌉ which expresses the property of vertices x, y to be connected by a chain of length at most i: Di (x, y) = ∃v (Di/2 (x, v) ∧ Di/2 (y, v)) if i is even and Di (x, y) = ∃v (D(i−1)/2 (x, v) ∧ D(i+1)/2 (y, v)) if i is odd. Therefore, the quantifier depth of Rz (a, b) equals k − 12. For any vertices a, b such that

 

˜ k (a, b) := D∗k−13 (a, b) ∧ ¬ ∃x1 ∃x2 ∃x3 D 2 

× ∨1≤i<2k−13 −1 (D1 (a, x1 ) ∧ D1 (a, x2 ) ∧ Di (x1 , x3 ) ∧ Di (x2 , x3 ) ∧ D2k−13 −1−i (x3 , b)) ∗

∗

∗

∗

∗



holds consider the set U (a, b) of all vertices z such that a ∼ z and D∗2k−13 −1 (z , b). Obviously, the property z ∈ U (a, b) can be expressed by a first-order formula with quantifier depth k − 13. Finally, the target first-order property L is defined by the first-order formula with quantifier depth k:

∃x∃y [D˜ k (x, y) ∧ (∀a∀b (D˜ k (a, b) ⇒ [(U (a, b) is P-smaller than U (x, y)) ∧ (U (x, y) is P-even)]))], where X is ‘‘P-smaller’’ than Y means that there exist z1 , z2 , z3 , z4 , z5 such that for all x ∈ X \ Y there is a unique y ∈ Y \ X with Rzi (x, y) for one of i ∈ {1, 2, 3, 4, 5} and for all y ∈ Y \ X there is at most one x ∈ X \ Y with this property, X is ‘‘P-even’’ means that there exist z1 , z2 , z3 such that for all x ∈ X there is a unique y ∈ X , y ̸= x, with Rzi (x, y) for one of i ∈ {1, 2, 3}. ˜ n ⊂ Ωn contain all graphs which satisfy the following property. Let Ω

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C If l < m/4 then for any pairs (a1 , b1 ), . . . , (al , bl ), which contain distinct vertices, and any set of vertices Γ , which contains the vertices a1 , b1 , . . . , al , bl , there exists a vertex z such that for any t ∈ {1, . . . , l} predicate Rz (at , bt ) is true and for any other pair of vertices (a, b) from Γ predicate Rz (a, b) is not true.

˜ n ) = 1. From Corollary 1 it follows that limn→∞ Pr(G(n, p) ∈ Ω ˜ n the property L can be replaced with the property L˜ which is defined as follows: Obviously, for any graph from Ω max

a,b: Dk (a,b)

|U (a, b)| is even.

Therefore, it remains to prove that for infinitely many m a limit limn→∞ Pr(G(n, p)  L˜ ) does not equal zero or one. Let m be even. From Theorems 16 and 15 it follows that for some c ∈ (0, 1)

˜ k (a, b) ∧ (|U (a, b)| = m)))) = c , lim Pr(G(n, p)  (∃a∃b (D

n→∞

but

˜ k (a, b) ∧ (|U (a, b)| = m − 1))] lim Pr(G(n, p)  ([∃a∃b (D

n→∞

∧ [∀a∀b (D˜ k (a, b) ⇒ (|U (a, b)| < m + 1))])) = 1. Thus, if m is even then limn→∞ Pr(G(n, p)  L˜ ) does not equal zero or one. Theorem 10 is proved. 5. Proof of Theorem 11 Consider a complete graph H with V (H ) = {v1 , . . . , vk−2 }. Let S be the set of all subsets of V (H ). Consider the pairs

(Kj , T ), j ∈ {0, 1, . . . , k − 2}, such that V (T ) = {u1 , . . . , uk−2 },

V (K0 ) = V (T ) ⊔ {uk−1 },

V (Kj ) = V (K0 ) ⊔ {u1 , . . . , uj },

E (K0 ) = E (T ) ⊔ {{u1 , uk−1 }, . . . , {uk−2 , uk−1 }}, E (Kj ) = E (K0 ) ⊔ {{u2 , u1 }, . . . , {uk−1 , u1 }, {u1 , u2 }, {u3 , u2 }, . . . , {uk−1 , u2 }, . . . ,

{u1 , uj }, . . . , {uj−1 , uj }, {uj+1 , uj }, . . . , {uk−1 , uj }}. For any S ∈ S define the graph HS in the following way. Let H (S ) be a graph which is obtained from H by renumbering its vertices: (i1 (S ), . . . , ik−2 (S )) → (1, . . . , k − 2), where S = {vi1 (S ) , . . . , vi|S | (S ) },

{v1 , . . . , vk−2 } \ S = {vi|S |+1 (S ) , . . . , vik−2 (S ) }.

Graph HS is a strict (K|S | , T )-extension HS of the graph H (S ). Let r ∈ {1, . . . , 2k−2 }. For any different S1 , . . . , Sr ∈ S denote HS1 ,...,Sr = HS1 ∪ · · · ∪ HSr (we define graphs HS in such a way that the intersection of any V (HSi ), V (HSi ) equals 1

2

{v1 , . . . , vk−2 }). Let us find the set of maximal densities of strictly balanced graphs HS1 ,...,Sr over all r ∈ {(k−2)2 , . . . , 2k−2 −1} and different S1 , . . . , Sr ∈ S \ ∅. Obviously, v(HS1 ,...,Sr ) = k − 2 + r + |S1 | + · · · + |Sr |, e(HS1 ,...,Sr ) =

(k − 2)(k − 3) 2

(10)

+ (k − 2)(r + |S1 | + · · · + |Sr |).

(11)

Therefore, 1/ρ max (HS1 ,...,Sr ) = 1/ρ(HS1 ,...,Sr ) =

1 k−2

+

k−1 . (k − 2)(k − 3 + 2(r + |S1 | + · · · + |Sr |))

Obviously, r + |S1 | + · · · + |Sr | takes on all the values from [(k − 2)3 , k2k−3 − 1] excluding at most 2k−2 values. Therefore, there are at least (k − 2)2k−3 − (k − 2)3 different values of 1/ρ max (HS1 ,...,Sr ). We are going to prove that for any k ≥ 7, r ∈ {(k − 2)2 , . . . , 2k−2 − 1} and different S1 , . . . , Sr ∈ S the point k−1 (12) (k − 2)(k − 3 + 2(r + |S1 | + · · · + |Sr |))    is in Sk1 and, therefore, prove that Sk1 ∩ 0, k−12.5  = Ω (k2k ). Let k ≥ 7, r ∈ {(k − 2)2 , . . . , 2k−2 − 1}, S1 , . . . , Sr ∈ S , α be defined by (12) and p = n−α . Let a first-order property L be defined by the following formula ϕ :

α=

1

k−2

+

ϕ = ∃x1 . . . ∃xk−2 (x1 ∼ x2 ) ∧ · · · ∧ (x1 ∼ xk−2 ) ∧ · · · ∧ (xk−3 ∼ xk−2 ) ∧ ϕS1 (x1 , . . . , xk−2 ) ∧ · · · ∧ ϕSr (x1 , . . . , xk−2 ),

J.H. Spencer, M.E. Zhukovskii / Discrete Mathematics 339 (2016) 1651–1664

1663

where

ϕS (x1 , . . . , xk−2 ) = ∃x k−1 (xk−1 ∼ x1 ) ∧ · · · ∧ (xk−1 ∼ xk−2 )    ∧ ϕ˜ t (x1 , . . . , xk−1 ) ∧  t ∈{i1 (S ),...,i|S | (S )}

 

ϕˆ t (x1 , . . . , xk−1 ) ,

t ̸∈{i1 (S ),...,i|S | (S )}

ϕ˜ t (x1 , . . . , xk−1 ) = ∃xk (xk ∼ x1 ) ∧ · · · ∧ (xk ∼ xt −1 ) ∧ (xk  xt ) ∧ (xk ∼ xt +1 ) ∧ · · · ∧ (xk ∼ xk−1 ), ϕˆ t (x1 , . . . , xk−1 ) = ∀xk (¬((xk ∼ x1 ) ∧ · · · ∧ (xk ∼ xt −1 ) ∧ (xk ∼ xt +1 ) ∧ · · · ∧ (xk ∼ xk−1 ))). Quantifier depths of formulae ϕ˜ t (x1 , . . . , xk−1 ), ϕˆ t (x1 , . . . , xk−1 ) equal 1. Therefore, quantifier depths of formulae ϕS (x1 , . . . , xk−2 ) equal 2. Finally, quantifier depth of formula ϕ equals k. Let a graph G ∈ Ωn follow L. Then there exist vertices x1 , . . . , xk−2 ∈ V (G) such that for any S ∈ {S1 , . . . , Sr } there is a strict (HS , H )-extension GS of G|{x1 ,...,xk−2 } (S ). Denote by xS1 , . . . , xSr the vertices of GS1 , . . . , GSr which are adjacent to each of x1 , . . . , xk−2 . As S1 , . . . , Sr are pairwise different, any two vertices of xS1 , . . . , xSr do not coincide. Moreover, for any distinct i1 , i2 ∈ {1, . . . , r } any vertex x ∈ V (GSi ) \ {x1 , . . . , xk−2 , xSi } do not coincide with xSi , because xSi is adjacent to each of 1 1 2 2 x1 , . . . , xk−2 and x is adjacent to exactly k−3 of them. Suppose that for any distinct i1 , i2 ∈ {1, . . . , r } any vertices x and y from V (GSi ) \ {x1 , . . . , xk−2 , xSi } and V (GSi ) \ {x1 , . . . , xk−2 , xSi } respectively do not coincide. Denote the union of GS1 , . . . , GSr 2 2 1 1 by G1 . If there are distinct i1 , i2 ∈ {1, . . . , r } such that the sets V (GSi ) \ {x1 , . . . , xk−2 , xSi }, V (GSi ) \ {x1 , . . . , xk−2 , xSi } 1 1 2 2 have common elements, then denote the union of GS1 , . . . , GSr by G2 . Let us prove that ρ(G1 ) < ρ(G2 ). Obviously, numbers of vertices and edges of G1 are defined by Eqs. (10) and (11). Moreover, there exists a natural number x such that v(G2 ) = v(G1 ) − x, e(G2 ) = e(G1 ) − x(k − 2) + x. Obviously, inequality e(G1 )

<

v(G1 )

e(G1 ) − x(k − 3)

v(G1 ) − x

holds when ρ(G1 ) > k − 3. The last inequality follows from 1/ρ(G1 ) <

1 k−2

+

k−1

(k − 2)(k − 3 + 2(k − 2) ) 2

=

1 k − 2.5

if r ≥ (k − 2) . From Theorem 15 it follows that a.a.s. in G(n, p) there is no copy of G2 . Moreover, from Theorem 16 it follows that there exists a number c ∈ (0, 1) such that limn→∞ Pr(G(n, p) ⊃ G1 ) = c. Therefore, limn→∞ Pr(G(n, p)  L) ≤ c. Note that if G ⊃ G1 and G1 is (K0 , T )-maximal in G, then G follows L. By Theorem 15 a.a.s. any copy of G1 in G(n, p) is (K , T )-maximal in G(n, p) (as any (K , T )-extension of its subgraph creates a graph with density greater than 1/ρ ). Therefore, limn→∞ Pr(G(n, p)  L) ≥ c. Finally, we have 2

lim Pr(G(n, p)  L) = c .

n→∞

Theorem 11 is proved. 6. Proof of Theorem 12 Theorem 11 implies inequalities k1 ≥ k2 ≥ 4. Let us prove that there exists a property L with quantifier depth 12 such that S 1 (L) is infinite. Let m ∈ N and α = 12 + 12m3+2 , p = n−α . Given a vertex z, consider the following predicate Rz (a, b) = (∃v1 ∃v2 ((v1 ∼ a) ∧ (v1 ∼ z ) ∧ (v2 ∼ v1 ) ∧ (v2 ∼ b))) and the property L, which is expressed by the sentence max

x 1 ,x 2 : x 1 ∼ x 2

|N (x1 , x2 )| is even.

˜ n ⊂ Ωn contain all graphs which satisfy the following properties. For any natural l, l ≤ m, and any pairs Let Ω (a1 , b1 ), . . . , (al , bl ), which contain distinct vertices, there exists a vertex z such that for any t ∈ {1, . . . , l} the predicate Rz (at , bt ) is true and for any other pair of vertices (a, b) predicate Rz (a, b) is not true. Moreover, there are no strictly balanced subgraphs with at most 3m + 2 vertices and density which is greater than 1/α . From Theorem 15 and Corollary 1 it follows ˜ n ) = 1. We apply this equality to prove the following statement. that limn→∞ Pr(G(n, p) ∈ Ω Lemma 3. There is a first order property L˜ which is expressed by formula ϕ with quantifier depth 12 such that lim (Pr(G(n, p)  L) − Pr(G(n, p)  L˜ )) = 0.

n→∞

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J.H. Spencer, M.E. Zhukovskii / Discrete Mathematics 339 (2016) 1651–1664

Proof. Consider the following first-order formula ϕ with quantifier depth 12:

∃x1 ∃x2 ((∀˜x1 ∀˜x2 N (˜x1 , x˜ 2 ) ≤3 N (x1 , x2 )) ∧ (N (x1 , x2 ) is 2-even)), where

 (A ≤ B) = 3













3 

∃z1 ∃z2 ∃z3 ∀x (x ∈ A \ B) ⇒ ∃y (y ∈ B \ A) ∧ Rzi (x, y) i=1        3 3   , Rzi (x, y˜ ) ∧ (x ∈ B \ A) ⇒ ¬ ∃y∃˜y (y ̸= y˜ ) ∧ Rzi (x, y) ∧ i=1

i=1

(N (x1 , x2 ) is 2-even) = (∃z1 ∃z2 ∀x ∈ N (x1 , x2 ) ∃y ∈ N (x1 , x2 ) × ((x ̸= y) ∧ [Rz1 (x, y) ∨ Rz2 (x, y) ∧ (∀˜y ∈ N (x1 , x2 ) ((y ̸= y˜ ) ⇒ (¬(Rz1 (x, y˜ ) ∨ Rz2 (x, y˜ )))))])). ˜ n and G  L. Then for any vertices x1 , x2 we get |N (x1 , x2 )| ≤ 3m. Therefore, from the definition of Ω ˜ n it follows Let G ∈ Ω ˜ that G  L. ˜ n ) = 1, Lemma 3 is proved. Obviously, L˜ implies L. As limn→∞ Pr(G(n, p) ∈ Ω From Theorem 15 it follows that a.a.s. in G(n, p) there are no vertices x1 , x2 such that x1 ∼ x2 and |N (x1 , x2 )| ≥ 3m + 1. Moreover, from Theorem 16 it follows that there exists a number c = c (m) ∈ (0, 1) such that in G(n, p) there are adjacent vertices x1 , x2 with |N (x1 , x2 )| = 3m with limiting probability c. Therefore, for any even m Lemma 3 implies 1 ′ limn→∞ Pr(G(n, p)  L˜ ) = limn→∞ Pr(G(n, p)  L) = c. Thus, 1/2 ∈ (S12 ). Finally, from the proof of Theorem 10 (see Section 4.5) it follows that for any k ≥ 10 there exists a first order formula with 1 is infinite. Theorem 12 is quantifier depth k, which expresses the property L with the limit point k−1 7 in S 1 (L). Therefore, S10 proved. Acknowledgments This work was carried out with the support of the Russian Foundation for Basic Research Grant No. 13-01-00612 and by the Grant No. 15-01-03530, grant of the Russian President MK-2184.2014.1. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

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