Extension of the Zero-one k-law

M.E. Zhukovskii

Dept. of Probability TheoryMSU

Moscow, Russia

Abstract

In this work we find new set of α such that the random graph G(N,N−α) obeyszero-one k-law.

Keywords: Random graph, zero-one law, quantifier depth.

Acknowledgement. This research was supported by the grant 12-01-00683of the Russian Foundation for Basic Research and grant MD-6277.2013.1.

1 Zero-one laws

We study asymptotical behaviour of the probabilities of first-order propertiesfor Erdos–Renyi random graphs G(N, p).

It was proved in 1969 by Y.V. Glebskii, D.I. Kogan, M.I. Liagonkii andV.A. Talanov in [1] (and independently in 1976 by R.Fagin in [2]) that forany first order property L either “almost all” graphs satisfy this property asN tends to infinity or “almost all” graphs don’t satisfy the property. In otherwords, if p doesn’t depend on N , then for any first-order property L eitherthe random graph satisfies the property L almost surely or it doesn’t satisfy(in such cases the random graph is said to obey zero-one law (see [1]–[7])).

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Moreover, the following statement holds.

Theorem 1 (Y.V. Glebskii, D.I. Kogan, M.I. Liagonkii, V.A. Talanov,1969, [1]; R.Fagin, 1976, [2]). Let a function p = p(N) satisfy the followingproperty: min(p, 1− p)Nα →∞ as N →∞ for any α > 0. Then the randomgraph G(N, p) obeys the zero-one law.

In this work we consider the probabilities p = p(N), where p(N) = N−α,N ∈ N, for α ∈ (0, 1). The zero-one law for such probabilities was proved byS. Shelah and J.H. Spencer (see [3]).

Theorem 2 (S. Shelah, J.H. Spencer, 1988, [3]). Let p(N) = N−α and αbe an irrational number such that 0 < α < 1. Then the random graph G(N, p)obeys the zero-one law.

When α ∈ (0, 1) is rational the zero-one law in ordinary sense for thesegraphs doesn’t hold (see [3]–[5]).

Until this moment we have considered the class of all first-order propertiesexpressed by the sentences of a finite length. The formulae containing aninfinite number of conjunctions and disjunctions were also studied. In 1997M. McArtur (see [6]) considered such formulae with quantifier depth boundedby a fixed number. In her work zero-one laws for the random graph G(N, p)with p = N−α and some rational α from (0, 1) were obtained.

Thus the random graph G(N, p) doesn’t obey zero-one law for the prop-erties from the class of all first-order properties expressed by the sentences ofa finite length if p = N−α and α from (0, 1) is rational. However for somerational α it obeys zero-one law when the sentences can be of an infinite lengthand quantifier depth is bounded. So it is natural to consider sentences of afinite length with bounded quantifier depth. The following question is arisen.Which rational α “force” the random graph G(N,N−α) to obey zero-one law(of course, if the random graph obeys zero-one from the work of McArtur,then it obeys “finite law”).

Let k be a positive integer. Denote by Lk the class of the first-order prop-erties of graphs defined by formulae with quantifier depth bounded by thenumber k (the sentences are of a finite length). Let us say that the randomgraph obeys zero-one k-law, if for any first-order property L ∈ Lk either therandom graph satisfies the property L almost surely or it doesn’t satisfy. In2012 we prove [7] that G(N,N−α), α ∈ (0, 1/(k − 2)), obeys zero-one k-law.

M.E. Zhukovskii / Electronic Notes in Discrete Mathematics 43 (2013) 263–269264

Theorem 3. (M.E. Zhukovskii, 2012, [7]). Let k ∈ N, k > 2. Letp = N−α, 0 < α < 1

k−2 . Then the random graph G(N, p) obeys the zero-one

k-law. If α = 1k−2 , then the random graph G(N, p) doesn’t obey the zero-one

k-law.

Recently we extend this law. Let us state this extended zero-one k-law.

Theorem 4. Let k > 3 be an arbitrary natural number. Let Q be the setof all positive rational numbers with numerator at most 2k−1. The randomgraph G(N,N−α) obeys zero-one k-law if α = 1− 1

2k−1+β, β ∈ (0,∞) \ Q.

To prove our result we employ theorems on estimates for the number ofextensions of small subgraphs in the random graph (see [4], [8]–[10]). Such anapproach was first used by J.H. Spencer and S. Shelah in 1988 (see [3], [4]).We also used our recent results [9].

2 Ideas of proof

The proof of Theorem 4 is based on the Ehrenfeucht game. Let us recall (see[4]–[7], [11]) this game (the game EHR(G,H, k) with two graphs G and H,two players (Spoiler and Duplicator) and a fixed number of rounds k). Atthe ν-th round (1 ≤ ν ≤ k) Spoiler chooses either a vertex xjν ∈ V (G) ora vertex yj′ν ∈ V (H). Duplicator chooses a vertex of the other graph. IfSpoiler chooses at the μ-th round the vertex xjμ ∈ V (G), jμ = jν (ν < μ),then Duplicator must choose yj′ν ∈ V (H). If at this round Spoiler choosesa vertex xjμ ∈ V (G), jμ /∈ {j1, ..., jμ−1}, then Duplicator must choose a ver-tex yj′μ ∈ V (H), j′μ /∈ {j′1, ..., j′μ−1}. If Duplicator cannot do it then Spoilerwins the game. At the end of the game the vertices xj1 , ..., xjk ∈ V (G);yj′1 , ..., yj′k ∈ V (H) are chosen. Let xh1 , ..., xhl

and yh′1 , ..., yh′l , l ≤ k, be dif-ferent vertices from the sets {xj1 , ..., xjk} and {yj′1 , ..., yj′k}. Duplicator wins ifand only if G|{xh1

,...,xhl} ∼= H|{yh′1 ,...,yh′l}. Let PN,p, PM,p be the distributions of

G(N, p), G(M, p). Denote PN,p(GN)PM,p(GM) by PN,M,p(GN ,GM) for any twoundirected graphs without loops and multiple edges GN on N vertices and GMon M vertices. The following statement holds (see [4], [5], [7]).

M.E. Zhukovskii / Electronic Notes in Discrete Mathematics 43 (2013) 263–269 265

Lemma 1. The random graph G(N, p) obeys the zero-one k-law if andonly if

limN,M→∞

PN,M,p (Duplicator wins the game EHR(G,H, k)) = 1.

If we want to prove the validity of the zero-one k-law, then we need to findthe winning strategy of Duplicator. Therefore, at each round we need to finda vertex that extends chosen vertices in a necessary way. Therefore, we needsome statements about the properties of extension in the random graph.

Let us consider graphs H with V (H) = {x1, ..., xk} and G with V (G) =

{x1, ..., xl}. Let H ⊂ G. A graph G with V (G) = {x1, ..., xl} is called a

(G,H)–extension of a graph H with V (H) = {x1, ..., xk}, if H ⊂ G and

{xi, xj} ∈ E(G) \ E(H)⇒ {xi, xj} ∈ E(G) \ E(H). If relation

{xi, xj} ∈ E(G) \ E(H)⇔

⇔ {xi, xj} ∈ E(G) \ E(H)

holds, then the extension is called strict.

Let α > 0 be a fixed number. Set v(G,H) = |V (G) \ V (H)|, e(G,H) =|E(G)\E(H)|. Denote f(G,H) = v(G,H)−α ·e(G,H). Moreover, denote thenumber of vertices in a graph G by v(G) and the number of edges by e(G).If for any such a graph S that H ⊂ S ⊆ G the inequality f(S,H) > 0 holds,then the pair (G,H) is called α-safe (see [4], [5], [8]). If for any such a graphS that H ⊆ S ⊂ G, the inequality f(G,S) < 0 holds, then the pair (G,H)is called α-rigid (see [4], [5]). If for any such graph S that H ⊂ S ⊂ G, theinequality f(S,H) > 0 holds, but f(G,H) = 0, then we call the pair (G,H)

α-neutral. Let H ⊂ G ⊂ Γ, T ⊂ K and |V (T )| ≤ |V (G)|. Let us call the

pair (G, H) (K,T )-maximal in Γ, if for any such subgraph T of the graph G

that |V (T )| = |V (T )| and T ∩ H �= T there doesn’t exist such a strict (K,T )–

extension K in Γ \ (G \ T ) that each vertex from V (K) \V (T ) is not adjacent

to each vertex from V (G) \ V (T ).

We are going to state some results on the estimates of the number ofextensions. Let us consider G(N,N−α) and some vertices x1, ..., xk from VN .Let a pair (G,H) be α-safe. Let V (H) = {x1, ..., xk}, V (G) = {x1, ..., xl}.Denote the number of (G,H)–extensions of G|{x1,...,xk} by N(G,H)(x1, ..., xk).In 1990 J.H. Spencer proved (see [4], [8]) that almost surely for any vertices

M.E. Zhukovskii / Electronic Notes in Discrete Mathematics 43 (2013) 263–269266

x1, ..., xk as N tends to infinity N(G,H)(x1, ..., xk) tend to infinity also and therelation

N(G,H)(x1, ..., xk) ∼ EN,pN(G,H)(x1, ..., xk)

is valid. Let Σneutral(r) be the set of all α-neutral pairs (Ki, Ti) where

|V (Ti)| ≤ |V (G)|, |V (Ki) \ V (Ti)| ≤ r. Let Σrigid(r) be the set of allα-rigid pairs (Ki, Ti), where |V (Ti)| ≤ |V (G)|, |V (Ki) \ V (Ti)| ≤ r. De-

note by Nneutral(G,H),r (x1, ..., xk) (by N

rigid(G,H),r(x1, ..., xk)) the number of all strict

(G,H)–extensions G of the graph H = G|{x1,...,xk} such that for each pair

(Ki, Ti) ∈ Σneutral(r) (pair (Ki, Ti) ∈ Σrigid(r)) the pair (G, H) is (Ki, Ti)-maximal in G. For these random variables a result similar to the theorem ofSpencer holds. Let us notice that this result is a corollary of our work [9] andones by J.H. Spencer and S. Shelah (see [3], [4]).

Theorem 4 (J.H. Spencer, S. Shelah, 1988, [3]; M.E. Zhukovskii, 2012,[9]). Almost surely for every vertices x1, ..., xk the following relations hold:

Nneutral(G,H),r (x1, ..., xk) ∼ EN,pN

neutral(G,H),r (x1, ..., xk) = Θ

(N f(G,H)

),

Nrigid(G,H),r(x1, ..., xk) ∼ EN,pN

rigid(G,H),r(x1, ..., xk) = Θ

(N f(G,H)

).

We exploit also some statements about the graphs of a specific type. Letus introduce some constructions.

Let m ∈ N, m ≥ 3, be a natural number. Let H,G be graphs such thatG ⊃ H. We call G a m–extension of H if one of the following properties holds.

— There exists a vertex x1 in V (G) such that V (G) = V (H) � {y11, ..., y1t1 , y21,..., y2t2}, E(G) = E(H)�{{x1, y

11}, {y11, y12}, ..., {y1t1−1, y1t1}, {y1t1 , y21}, {y21, y22},

..., {y2t2−1, y2t2}, {y2t2 , y1t1}}, where t1 + t2 ≤ m− 1, t1 ≥ 0, t2 ≥ 2, ρmax(G) <m

m−1 .

— There exist different vertices x1, x2 in G such that G = (V (H)�{y1, ..., yt},E(H) � {{x1, y1}, {y1, y2}, ..., {yt−1, yt}, {yt, x2}}), where t ≤ m − 1 andρmax(G) < m

m−1 .

Let us define a set Hm. Let x be a vertex. The graph ({x},∅) is in Hm.Let G = (V,E) ∈ Hm. The set Hm contains all pairwise non-isomorphic m–extensions of G. Note that for any G ∈ Hm either G = ({x},∅) or thereexists G1, ..., Gt such that G0 = ({x},∅) ⊂ G1 ⊂ ... ⊂ Gt = G, Gi �= Gi+1

for all i ∈ {0, 1, ..., t − 1} and the following property holds. Graph Gi isa m–extension of graph Gi−1 for all i ∈ {1, ..., t}. We call such collection

M.E. Zhukovskii / Electronic Notes in Discrete Mathematics 43 (2013) 263–269 267

G0, G1, ..., Gt a m–decomposition of G. We prove the following statement.

Lemma 2. Let G ∈ Hm and G0, G1, ..., Gt — its m–decomposition. Lett ≥ 2. Then there exist natural numbers a, b such that b ≤ m and ρmax(G) =1 + 1

m−1+b/a.

Let ρ ∈ (1,m/(m − 1)) be a real number. There exists a number η ∈ N

such that for any natural v > η and any graph G ∈ Hm on v vertices thereexists a subgraph in G with at most η vertices and density greater than ρ.

Exploiting Theorem 4 we prove that Duplicator can choose vertices inthe random graphs G(N, p) and G(M, p) according to the following strategy.Let vertices x1, ..., xs of G(N, p) and y1, ..., ys of G(M, p) be chosen and Spoilerchoose a vertex xs+1. Then with high probability in G(N, p) and G(M, p) thereexist graphs X1, ..., Xi(s+1) and Y1, ..., Yi(s+1) such that i = i(s+1) ≤ s+1 andthe following properties hold. The vertices x1, ..., xs+1 are in V (X1 ∪ ...∪Xi),the vertices y1, ..., ys are in V (Y1 ∪ ...∪ Yi). Graphs Xj and Yj are isomorphicfor each j in {1, ..., i}. For any j ∈ {1, ..., i} graph Xj is a union of some graphfrom H2k−m(j) with some tree. Numbers m(j) satisfy the following inequalities:1 = m(1) < m(2) < ... < m(i) ≤ s + 1. Let Duplicator choose the vertexϕ(xs+1), where ϕ : X1 ∪ ... ∪Xi → Y1 ∪ ... ∪ Yi is an isomorphism. Duplicatorwins with high probability as the graphs X1 ∪ ...∪Xi(k) and Y1 ∪ ...∪ Yi(k) areisomorphic.

References

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[2] R. Fagin, Probabilities in finite models, J. Symbolic Logic 41: 50-58, 1976.

[3] S. Shelah, J.H. Spencer, Zero-one laws for sparse random graphs, J. Amer.Math. Soc. 1: 97-115, 1988.

[4] N. Alon, J.H. Spencer, The probabilistic method, Second edition, Tel Aviv andNew York, John Wiley & Sons, INC, 2000.

[5] S. Janson, T. Luczak, A. Rucinski, Random Graphs, New York, Wiley, 2000.

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[6] M. McArtur, The asymptotic behavior of Lk∞,ω on sparse random graphs, Logicand Random Structures, 33: 53-63, 1997.

[7] M.E. Zhukovskii, Zero-One k-Law, Discrete Mathematics, 2012, 312: 1670–1688.

[8] J.H. Spencer, Counting extensions, J. Combinatorial Theory, Ser. A 55: 247-255, 1990.

[9] M.E. Zhukovskii, Estimation of the number of maximal extensions in therandom graph, Discrete Mathematics and Applications, 2012, 24(1): 79–107.

[10] B. Bollobas, Random Graphs, 2nd Edition, Cambridge University Press, 2001.

[11] A. Ehrenfeucht, An application of games to the completness problem forformalized theories, Warszawa, Fund. Math. 49: 121-149, 1960.

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