Finite asymptotic clusters of metric spaces

Let $(X, d)$ be an unbounded metric space and let $\tilde r=(r_n)_{n\in\mathbb N}$ be a sequence of positive real numbers tending to infinity. A pretangent space $\Omega_{\infty, \tilde r}^{X}$ to $(X, d)$ at infinity is a limit of the rescaling sequence $\left(X, \frac{1}{r_n}d\right).$ The set of all pretangent spaces $\Omega_{\infty, \tilde r}^{X}$ is called an asymptotic cluster of pretangent spaces. Such a cluster can be considered as a weighted graph $(G_{X, \tilde r}, \rho_{X})$ whose maximal cliques coincide with $\Omega_{\infty, \tilde r}^{X}$ and the weight $\rho_{X}$ is defined by metrics on $\Omega_{\infty, \tilde r}^{X}$. We describe the structure of metric spaces having finite asymptotic clusters of pretangent spaces and characterize the finite weighted graphs which are isomorphic to these clusters.


Introduction
Under an asymptotic cluster of metric spaces we mean the set of metric spaces which are the limits of rescaling metric spaces X, 1 rn d for r n tending to infinity. The Gromov-Hausdorff convergence and the asymptotic cones are most often used for construction of such limits. Both of these approaches are based on higher-order abstractions (see, for example, [19] for details), which makes them very powerful, but it does away the constructiveness. In this paper we use a more elementary, sequential approach for describing scaling limits of unbounded metric spaces at infinity.
Let (X, d) be a metric space and letr = (r n ) n∈N be a sequence of positive real numbers with lim n→∞ r n = ∞. In what followsr will be called a scaling sequence and the formula (x n ) n∈N ⊂ A will be mean that all elements of the sequence (x n ) n∈N belong to the set A. Definition 1.1. Two sequencesx = (x n ) n∈N ⊂ X andỹ = (y n ) n∈N ⊂ X are mutually stable with respect to the scaling sequencer = (r n ) n∈N if there is a finite limit lim n→∞ d(x n , y n ) r n :=dr(x,ỹ) =d(x,ỹ). (1.1) Let p ∈ X. Denote by Seq(X,r) the set of all sequencesx = (x n ) n∈N ⊂ X for which there is a finite limit lim n→∞ d(x n , p) r n :=dr(x) (1.2) and such that lim n→∞ d(x n , p) = ∞.
Definition 1.2. A set F ⊆ Seq(X,r) is self-stable if any twox,ỹ ∈ F are mutually stable. F is maximal self-stable if it is self-stable and, for arbitrarỹ y ∈ Seq(X,r), we have eitherỹ ∈ F or there isx ∈ F such thatx andỹ are not mutually stable.
The maximal self-stable subsets of Seq(X,r) will be denoted asX ∞,r . In particular, Seq(X,r), the self-stable subsets and the maximal self-stable subsets of Seq(X,r) are invariant w.r.t. the choosing a point p ∈ X in (1.2).
Every metric is a pseudometric. A pseudometric µ : Y × Y → R + is a metric if and only if, for all x, y ∈ Y, the equality µ(x, y) = 0 implies x = y.
Hence (X ∞,r ,d) is a pseudometric space. Now we are ready to define the main object of our research.
Definition 1.4. Let (X, d) be an unbounded metric space, letr be a scaling sequence and letX ∞,r be a maximal self-stable subset of Seq(X,r). The pretangent space to (X, d) (at infinity, with respect tor) is the metric identification of the pseudometric space (X ∞,r ,d).
Since the notion of pretangent space is basic for the paper, we recall the metric identification construction. Define a relation ≡ on Seq(X,r) as (x ≡ỹ) ⇔ dr (x,ỹ) = 0 . (1.4) The reflexivity and the symmetry of ≡ are evident. Letx,ỹ,z ∈ Seq(X,r) andx ≡ỹ, andỹ ≡z. Then the inequality lim sup n→∞ d(x n , z n ) r n ≤ lim n→∞ d(x n , y n ) r n + lim n→∞ d(y n , z n ) r n impliesx ≡z. Thus ≡ is an equivalence relation.
Write Ω X ∞,r for the set of equivalence classes generated by the restriction of ≡ on the setX ∞,r . Using general properties of pseudometric spaces we can prove (see, for example, [11]) that the function ρ : Ω X ∞,r × Ω X ∞,r → R with ρ(α, β) :=dr(x,ỹ),x ∈ α ∈ Ω X ∞,r ,ỹ ∈ β ∈ Ω X ∞,r , (1.5) is a well-defined metric on Ω X ∞,r . The metric identification of (X ∞,r ,d) is the metric space (Ω X ∞,r , ρ). Let us denote byX ∞ the set of all sequences (x n ) n∈N ⊂ X satisfying the limit relation lim n→∞ d(x n , p) = ∞ with p ∈ X. It is clear that Seq(X,r) ⊆X ∞ holds for every scaling sequencer and for everyx ∈X ∞ , there exists a scaling sequencer such thatx ∈ Seq(X,r).
For every unbounded metric space (X, d) and every scaling sequencer define the subsetX 0 ∞,r of the set Seq(X,r) by the rule: (z n ) n∈N ∈X 0 ∞,r ⇔ (z n ) n∈N ∈X ∞ and lim n→∞ d(z n , p) r n = 0 , (1. 6) where p is a point of X.
Below we collect together some basic properties of the setX 0 ∞,r .
Proposition 1.5. Let (X, d) be an unbounded metric space and letr be a scaling sequence. Then the following statements hold.
(iii) If F ⊆ Seq(X,r) is self-stable, thenX 0 ∞,r ∪F is also a self-stable subset of Seq(X,r).
(vii) Denote by Ω X ∞,r the set of all pretangent to X at infinity (with respect tor) spaces. Then the membership

holds.
A simple proof is omitted here.
Remark 1.6. The setX 0 ∞,r is invariant under replacing of p ∈ X by an arbitrary point b ∈ X in (1.6). Lemma 1.7. Let (X, d) be an unbounded metric space, p ∈ X andỹ ∈X ∞ , letr be a scaling sequence and letX ∞,r be a maximal self-stable set. Ifỹ and x are mutually stable for everyx ∈X ∞,r , thenỹ ∈X ∞,r .
Proof. Supposeỹ andx are mutually stable for everyx ∈X ∞,r . To provẽ y ∈X ∞,r it suffices to show that there is a finite limit lim n→∞ d(yn,p) rn that follows from statements (v ) and (vi ) of Proposition 1.5.
Lemma 1.8. Let (X, d) be an unbounded metric space and letr be a scaling sequence. Ifx,ỹ,t ∈X ∞ such thatx andỹ are mutually stable with respect tor anddr(x,t) = 0, thenỹ andt are mutually stable with respect tor.
Proof. The statement follows from the equalityd r (x,t) = 0 and the inequalities The setX 0 ∞,r is a common distinguished point of all pretangent spaces Ω X ∞,r (with given scaling sequencer). We will consider the pretangent spaces to (X, d) at infinity as the triples (Ω X ∞,r , ρ, ν 0 ), where ρ is defined by (1.5) and ν 0 :=X 0 ∞,r . The point ν 0 can be informally described as follows. The points of pretangent space Ω X ∞,r are infinitely removed from the initial space (X, d), but Ω X ∞,r contains a unique point ν 0 which is close to (X, d) as much as possible. Example 1.9. Let (x n ) n∈N ⊂ (0, ∞) be an increasing sequence andr = (r n ) n∈N be a scaling sequence such that for every n ∈ N. Define a metric space (X, d) as and d(x, y) := |x − y| for all x, y ∈ X. It follows from (1.7) and (1.8) that, for every n ∈ N, we have either Consequently the equalitỹ dr(ỹ) = 0 holds for everyỹ ∈ Seq(X,r), i.e., In conclusion of this introduction we note that there exist other techniques which allow to investigate the asymptotic properties of metric spaces at infinity. As examples, we mention only the Gromov product which can be used to define a metric structure on the boundaries of hyperbolic spaces [4], [20], the balleans theory [18] and the Wijsman convergence [13], [23], [24].

The cluster of pretangent spaces
In this section, using some elements of the graph theory, we introduce the concept of cluster of pretangent spaces which will allow us to describe the relationships between these spaces.
Recall that a graph G is an ordered pair (V, E) consisting of a nonempty set V = V (G) and a set E = E(G) of unordered pairs of distinct elements of V (G). The elements of V and E are called the vertices and, respectively, the edges of G. Thus all our graph are simple and loopless. In what follows we mainly use the terminology from [2]. In particular, we say that vertices x and y of a graph G are adjacent if {x, y} ∈ E(G).
Let (X, d) be an unbounded metric space and letr be a scaling sequence. Let us consider the graph G X,r with the vertex set V (G X,r ) consisting of the equivalence classes generated by the relation ≡ on Seq(X,r) (see (1.4)) and the edge set E(G X,r ) defined by the rule: Recall that a clique in a graph G = (V, E) is a set C ⊆ V such that every two distinct vertices of C are adjacent. A maximal clique is a clique C 1 such that the inclusion C 1 ⊆ C implies the equality C 1 = C for every clique C in G.
Theorem 2.1. Let (X, d) be an unbounded metric space and letr be a scaling sequence. A set C ⊆ V (G X,r ) is a maximal clique in G X,r if and only if there is a pretangent spaces (Ω X ∞,r , ρ) such that C = Ω X ∞,r . Proof. Lemma 1.7 and Lemma 1.8 imply the equality for everyỹ ∈X ∞,r and everyX ∞,r ⊆ Seq(X,r). Since, for everyỹ ∈ Seq(X,r), there isX ∞,r such thatX ∞,r ∋ỹ, equality (2.1) implies where Ω X ∞,r is the set of all spaces which are pretangent to X at infinity with respect tor. Now the theorem follows from the definitions of the pretangent spaces and the maximal cliques. Theorem 2.1 gives some grounds for calling the graph G X,r a cluster of pretangent spaces to (X, d) at infinity.
Recall that a vertex v of a graph G = (V, E) is dominating if {u, v} ∈ E holds for all u ∈ V \ {v}. Statement (vii ) of Proposition 1.5 gives us the following fact.
Proposition 2.2. Let (X, d) be an unbounded metric space and letr be a scaling sequence. Then the vertex ν 0 =X 0 ∞,r is a dominating vertex of G X,r . If G = (V, E) is a simple graph and r ∈ V is a distinguished vertex of G, then we will say that G is a rooted graph with the root r and write G = G(r).
Now we recall the definition of isomorphic rooted graphs.
holds for all u, v ∈ V (G 1 ). The rooted graphs G 1 and G 2 are isomorphic if there exists an isomorphism f : The isomorphism of rooted graphs is a special case of the graph homomorphisms whose theory is a relatively new but very promising branch of the graph theory. See the book of Pavel Hell and Jaroslav Nešetřil [10].
If (X, d) is an unbounded metric space andr is a scaling sequence, then we will consider the cluster G X,r as a rooted graph with the root ν 0 =X 0 ∞,r and write G X,r = G X,r (ν 0 ). Problem 2.4. Describe the rooted graphs which are isomorphic to the rooted clusters of pretangent spaces.
Remark 2.5. Using Proposition 2.2 we can prove that if T (r) is a nontrivial rooted tree and this tree is isomorphic to a rooted cluster G X,r (ν 0 ), then T (r) is a star. Thus the class of rooted clusters of pretangent spaces is a proper subclass of the class of all rooted graphs.
The following, important for us, notion is a weighted graph, i.e., a simple graph G = (V, E) together with a weight w : E → R + . Let us define a weight ρ X on the edge set of G X,r as: there is a pretangent space (Ω X ∞,r , ρ) such that u, v ∈ Ω X ∞,r , we have ρ X ({u, v}) = ρ(u, v).
(2.5) Definition 2.6. Let G i = G i (w i , r i ) be weighted rooted graphs with the roots r i and the weights w i : of the rooted graphs G 1 (r 1 ) and G 2 (r 2 ) is an isomorphism of the weighted rooted graphs G 1 (w 1 , r 1 ) and G 2 (w 2 , r 2 ) if the equality holds for every {u, v} ∈ E(G 1 ). Two weighted rooted graphs are isomorphic if there is an isomorphism of these graphs.
Problem 2.7. Describe the weighted rooted graphs which are isomorphic to the weighed rooted clusters of pretangent spaces.
Problem 2.4, that was formulated above, is a weak version of Problem 2.7. For the finite graphs, both those problems will be solved in the next section of the paper.
The solution of these problems is based on the following fact: "The weighted clusters G X,r (ρ X ) are metrizable".
Recall that a weighted graph holds for every {u, v} ∈ E(G). Similarly, G(w) is pseudometrizable if there is a pseudometric δ : V (G)×V (G) → R + such that (2.7) holds for every {u, v} ∈ E(G). In this case we say that G(w) is metrizable (pseudometrizable) by the metric (pseudometric) δ. Let G(w) be a connected weighted graph and let u, v be distinct vertices of G. Let us denote by P u,v the set of all paths joining u and v in G. Write where w(P ) := e∈P w(e). The function d * w is a pseudometric on the set . This pseudometric will be termed as the weighted shortest-path pseudometric. It coincides with the usual path metric if w(e) = 1 for every e ∈ E(G).
The following lemma is a simplified version of Proposition 2.1 from [8].
Lemma 2.8. Let G = G(w) be a connected weighted graph. The following statements are equivalent.
(i) The graph G(w) is pseudometrizable.
The next lemma follows directly from Lemma 2.8, the triangle inequality and the definition of the shortest-path pseudometric.
Proposition 2.10. Let (X, d) be an unbounded metric space andr be a scaling sequence. Then the shortest-path pseudometric d * ρ X is a metric and the weighted cluster G X,r (ρ X ) is metrizable by d * ρ X . Proof. Lemma 2.9 and Lemma 2.8 imply that the shortest-path pseudometric where (x n ) n∈N ∈ u and (y n ) n∈N ∈ v. It follows directly from the definitions of G X,r and ρ X that holds for every {u, v} ∈ E(G X,r ). As in (1.5) we can see that ∆ is welldefined on V (G X,r ) × V (G X,r ). We claim that ∆ is a metric on V (G X,r ). The inequalities forx ∈ u andỹ ∈ v. Thusx ≡ỹ holds (see (1.4)). It implies that u = v.

The metric spaces with finite clusters of pretangent spaces
In this section we describe the unbounded metric spaces (X, d) having finite clusters G X,r for every scaling sequencer. Let p be a point of a metric space (X, d). Denote for r > 0 and k ≥ 1. The set S(p, r) is the sphere in (X, d) with the radius r and the center p. Analogously we can consider A(p, r, k) as an annulus in (X, d) "bounded" by the concentric spheres S(p, rk) and S(p, r k ). In particular, the annulus A(p, r, 1) coincides with the sphere S(p, r).
Theorem 3.1. Let (X, d) be an unbounded metric space, p ∈ X, and let n ≥ 2 be an integer number. Then the inequality holds for every scaling sequencer if and only if where r ∈ (0, ∞) and k ∈ [1, ∞) and the function F n : X n → R is defined as if (x 1 , . . . , x n ) = (p, . . . , p) and F n (p, . . . , p) := 0.
is continuous at the point 1 and Ψ(1) = 0 holds. In order to prove Theorem 3.1, it is necessary to find a connection between conditions (3.2) -(3.3) and the structure of the weighted rooted cluster G X,r (ρ X , ν 0 ). Theorem 2.1 and Theorem 4.3 from [1] imply the following lemma.
Lemma 3.4. Let (X, d) be an unbounded metric space and let n ≥ 2 be an integer number. The following statements are equivalent.
(i) The inequality |C| ≤ n holds for every clique C of each cluster G X,r .
(ii) Limit relation (3.2) holds for the function F n defined by equality (3.4).
Recall that, for given (X, d) andr, the weight ρ X is defined as: where ν 0 =X 0 ∞,r is the root of the cluster G X,r . By Proposition 2.2, ν 0 is a dominating vertex of G X,r . Hence ρ 0 is a well-defined function on V (G X,r ).
Recall also that an independent set I in a graph G is a subset of V (G) such that, for any two vertices in I, there is no edge connecting them.
The following lemma is an expanded version of Theorem 4.5 from [1].
Lemma 3.5. Let (X, d) be an unbounded metric space and p ∈ X. Then condition (3.3) from Theorem 3.1 holds if and only if the labeling ρ 0 : V (G X,r ) → R + is an injective function on V (G X,r ) for everyr. Moreover, if for givenr, there are two distinct vertices ν 1 , ν 2 ∈ V (G X,r ) and c ∈ R + with then there exists an independent set I ⊆ V (G X,r ) having the cardinality of the continuum, |I| = c, and such that holds for every v ∈ I.
Proof. Suppose condition (3.3) holds but there are a scaling sequencer and ν 1 , ν 2 ∈ V (G X,r ) and c ∈ R + such that ν 1 = ν 2 and Consequently, by the definition ofX 0 ∞,r , the statements Without loss of generality we may suppose that and lim n→∞ k n = 1. Since we have n and x 2 n for every n ∈ N. It follows from x 1 n , x 2 n ∈ A(p, R n , k n ) and lim n→∞ k n = 1 and (3.8) and (3.9) that lim n→∞ R n = ∞ and, for every k > 1, (See Remark 3.3.) It is easy to see that Ψ is increasing and Ψ(k) ≤ 2k holds for every k ∈ [1, ∞). Consequently, there is a finite limit

Moreover, condition (3.3) does not hold if and only if
and let b 1 ∈ (0, b). Then there are some sequencesx,ỹ ⊂ X and a sequencẽ r ⊂ (0, ∞) such that lim n→∞ r n = ∞ and x n , y n ∈ A(p, r n , k n ) (3.11) and hold for every n ∈ N. Statement (3.11) implies the inequalities for every n. Using (3.13) and (3.10) we obtaiñ Hence the labeling ρ 0 : V (G X,r ) → R + is not injective, contrary to our supposition. It still remains to find an independent set I ⊆ V (G X,r ) with |I| = c for G X,r having a non-injective labeling ρ 0 : V (G X,r ) → R + .
Suppose there existr and ν 1 , ν 2 ∈ V (G X,r ) such that ν 1 = ν 2 and ρ 0 ( and Let N e be an infinite subset of N such that N \ N e is also infinite and We can consider a relation ≍ on the set 2 Ne of all subsets of N e defined by the rule: A ≍ B, if and only if the set The set of all finite subsets of N e is countable. Consequently equality (3.16) implies Let N ⊆ 2 Ne be a set such that: holds for all N 1 , N 2 ∈ N . It follows from (3.17) that |N | = c. For every N ∈ N define the sequencẽ x(N) = (x n (N)) n∈N as Recall that (x 1 n ) n∈N , (x 2 n ) n∈N ∈ Seq(X,r) satisfy (3.14) and (3.15). It follows from (3.14) and (3.15) that for every N ∈ N . Thusx(N) ∈ Seq(X,r). Let N 1 and N 2 be distinct elements of N . Then, by (3.19), the equality holds for every n ∈ N 1 △ N 2 . Using (3.15) and the definition of ≍ we see that the set N 1 △ N 2 is infinite for all distinct N 1 , N 2 ∈ N . Consequently, we have  The first inequality in (3.21) is an independent set in G X,r and |I| = c holds. To complete the proof note that (3.20) implies (3.6) for every v ∈ I.
Remark 3.6. The existence of continuum many sets A γ ⊆ N satisfying, for all distinct γ 1 and γ 2 , the equalities  holds for all cliques C ⊆ V (G X,r ).

Structural characteristic of finite G X,r
Our next goal is the structural characteristic of the finite, weighted, rooted graphs which are isomorphic to the weighted rooted clusters of pretangent spaces. This characteristic will be based on the concept of a cycle. Recall that a graph C is a subgraph of the graph G, C ⊆ G, if A finite graph C is a cycle in a graph G if C ⊆ G and |V (C)| ≥ 3 and there exists a numbering (v 1 , . . . , v n ) of V (C) such that For a weighted graph G = G(w), the length of a cycle C ⊆ G is defined as If V (C) = (v 1 , . . . , v n ) and (4.1) holds, then we have We need several lemmas. Let (X, d) be an unbounded metric space,r = (r n ) n∈N be a scaling sequence andr ′ = (r n k ) k∈N be a subsequence ofr. Denote by Φr′ the mapping from Seq(X,r) to Seq(X,r ′ ) with It is clear that anddr(x) =dr′(x ′ ) for everyx ∈ Seq(X,r). Consequently, there is a mapping such that the diagram Φr′ πr πr′ Em ′ is a weight preserving homomorphism of G 1 (w 1 , r 1 ) and G 2 (w 2 , r 2 ) if the following statements hold: A weight preserving monomorphism of the graphs G 1 (w 1 , r 1 ) and G 2 (w 2 , r 2 ) is an injective and weight preserving homomorphism of these graphs.
Hence (y n ) n∈N ∈ Seq(X,r). Moreover, we have Consequently, With no loss of generality suppose that lim sup n→∞ d(x 1 n , y n ) r n > 0. Write ν 3 := πr(ỹ) (see diagram 4.5). Inequality (4.8) implies that ν 1 = ν 2 . Now, forr ′ = (r n k ) k∈N with n k = 2k, equality (4.7) shows that Thus Em ′ is not a monomorphism.  Let (X, d) be an infinite metric space, letr be a scaling sequence and let u * , v * be distinct non adjacent vertices of G X,r . If G X,r is finite, then there are two metrics d 1 , d 2 ∈ M(ρ X ) such that where (x n ) n∈N ∈ u * and (y n ) n∈N ∈ v * . Letr ′ 1 = r n 1,k k∈N andr ′ 2 = r n 2,k k∈N be subsequences ofr satisfying the equalities lim k→∞ d(x n 1,k , y n 1,k ) r n 1,k = lim sup n→∞ d(x n , y n ) r n and lim k→∞ d(x n 2,k , y n 2,k ) r n 2,k = lim inf n→∞ d(x n , y n ) r n respectively. Suppose G X,r is finite. Then, by Lemma 3.5, the labeling ρ 0 : V (G X,r ) → R + is injective. Consequently, by Lemma 4.3, are the weight preserving monomorphisms (see diagram (4.5)). By Proposition 2.10 the weighted clusters G X,r ′ 1 and G X,r ′ 2 are metrizable by the corresponding shortest-path metrics d * and Em ′ 2 are weight preserving monomorphisms, the weighted cluster G X,r is metrizable by d 1 and d 2 . Moreover, (4.10) implies that d 1 (u * , v * ) = d 2 (u * , v * ). holds for every d ∈ M(w) and all distinct, non adjacent vertices µ, ν ∈ V (G), where (·) + is the positive part of (·). Conversely, if µ and ν are some distinct, non adjacent vertices of G and t is a positive real number satisfying the double inequality Proof. Let µ, ν ∈ V (G) be distinct and non adjacent and let d ∈ M(w).
Then the second inequality in (4.11) follows from Lemma 2.9. To prove the first inequality in (4.11) it suffices to show that the inequality holds for every path (x 1 , . . . , x n ) ⊆ G if x 1 = µ and x n = ν. When the left side of (4.13) is 0, then there is nothing to prove. In the opposite case, (4.13) can be written as that immediately follows from the triangle inequality. Suppose now that µ and ν are distinct, non adjacent vertices of G and t is a positive real number satisfying double inequality (4.12). We must find d ∈ M(w) such that d(µ, ν) = t. Let us consider the weighted grapĥ G =Ĝ(ŵ) with holds for every cycle C ⊆Ĝ. If C ⊆ G, then (4.14) holds because G(w) is metrizable. Let C G. Then {µ, ν} is an edge of the cycle C. There are two cases to consider: Let P be the path in C such that V (P ) = V (C) and {µ, ν} / ∈ E(P ). Then we evidently have P ∈ P µ,ν and (4.15) Consequently in the case when (i 1 ) holds, inequality (4.14) can be written as:  Since P ∈ P µ,ν and P ⊆ G, we have w(e).
The last inequality and the second inequality in (4.12) imply (4.16). It follows form (i 2 ) that max w(e). Using the first inequality in (4.12) and the membership P ∈ P µ,ν we obtain  This inequality, (4.15) and (4.17) imply (4.14).
Proof. Let µ and ν be distinct, non adjacent vertices of C and let e * be an edge of C such that w(e * ) = max Equality (4.18) implies that e * is the unique edge satisfying (4.19). For the cycle C, the set P µ,ν contains exactly two paths: P 1 with e * ∈ E(P 1 ) and P 2 with e * / ∈ E(P 2 ) (see Figure 1). It follows from (4.18) that C w(e). w(e).
Lemma 4.8. Let G = G(w) be a finite, connected, metrizable weighted graph and let µ, ν be distinct, non adjacent vertices of G. Then the following statements are equivalent: (ii) There is a cycle C ⊆ G such that µ, ν ∈ V (C) and w(e).
Since G is a finite graph, the last equality implies that w(e) > 0 (4.22) for some P 1 , P 2 ∈ P µ,ν .
Let we consider the graph P 1 ∪ P 2 , where P 1 , P 2 ∈ P µ,ν such that (4.22) holds. It is clear that w(e).

Moreover (4.22) implies the inequality
w(e). It suffices to show that P 1 ∪ P 2 is a cycle in G. Indeed, if P 1 ∪ P 2 is a cycle, then the converse inequality 2 max w(e).
To prove that P 1 ∪ P 2 is a cycle in G, we can consider the edge-deleted subgraph P 1,2 := P 1 ∪ P 2 − {e * } of the graph P 1 ∪ P 2 such that e * = {u * , v * } is the unique edge of P 1 ∪ P 2 with max It is clear that P 1,2 is connected. Consequently there is a path P 0 joining u * and v * in P 1,2 . Then C 0 := P 0 + e * is a cycle in P 1 ∪ P 2 . By Lemma 4.1 we have 2w(e * ) = 2 max w(e). Since C 0 ⊆ P 1 ∪ P 2 , the inequality holds. Inequalities (4.23), (4.24) and (4.25) imply the equality w(e).
Using the last equality and the inclusion C 0 ⊆ P 1 ∪ P 2 we see that C 0 = P 1 ∪ P 2 . Thus P 1 ∪ P 2 is a cycle in G.
We denote by F P C (Finite Pretangent Clusters) the class of all weighted rooted graphs G = G(w, r) for which |V (G)| < ∞ and there are an unbounded metric space (X, d) and a scaling sequencer such that G(w, r) and G X,r (ρ X , ν 0 ) are isomorphic as weighted rooted graphs. (See Definition 2.6.) If for a weighted rooted graph G(w, r) the root r is a dominating vertex, then we can define an analog w 0 of the labeling ρ 0 : V (G X,r ) → R + as follows: (4.26) The following theorem gives us a solution of Problem 2.7 for the finite graphs.  Proof. Let (X, d) be an infinite metric space andr be a scaling sequence for which there is an isomorphism of the weighted rooted graphs G(w, r) and G X,r (ρ X , ν 0 ). We must show that (i), (ii), and (iii) hold.
(i) By Proposition 2.2 the vertex ν 0 =X 0 ∞,r is a dominating vertex of G X,r . Since f is an isomorphism of rooted graphs, we have f (r) = ν 0 . Consequently, r is a dominating vertex of G. The graph G is finite by the condition. Hence G X,r is also finite. Now, using Lemma 3.5, we obtain that the labeling ρ 0 : V (G X,r ) → R + is injective. By the definition of w 0 (see (4.26)) the equality w 0 (v) = w({r, v})  Thus (ii) holds.
(iii) Suppose (iii) does not hold. Then there is a cycle C ⊆ G and some distinct vertices µ, ν ∈ V (C) such that and {µ, ν} / ∈ E(G). Since f : V (G) → V (G X,r ) is an isomorphism of weighted graphs, f (C) is a cycle in G X,r and f (µ), f (ν) ∈ V (f (C)) and Lemma 4.8 implies i.e., the equality holds for all d 1 , d 2 ∈ M(ρ X ). It follows from Lemma 4.4 that contrary to (4.31). Condition (iii) follows. Conversely, suppose that conditions (i), (ii) and (iii) hold for the weighted rooted graph G(w, r). We must find an unbounded metric space (X, d) and a scaling sequencer = (r n ) n∈N such that G(w, r) and G X,r (ρ X , ν 0 ) are isomorphic as weighted rooted graphs.
The set E un (G) (see Definition 4.6) is empty. To see it suppose {µ, ν} ∈ E un (G). Since w(e) > 0 for every e ∈ E(G), condition (ii) and Lemma 4.1 imply that G(w) is metrizable. By Lemma 4.8, there is a cycle C ⊆ G such that holds and µ, ν ∈ V (C). It follows from condition (iii) that V (C) is a clique in G. Hence {µ, ν} ∈ E(G). The last statement contradicts the definition of E un (G).
Let G be the complement of G, i.e., G is the graph whose vertex set is V (G) and whose edges are the pairs of nonadjacent vertices of G (see [2,Definition 1.1.17]). Since E un (G) = ∅, for every e = {u, v} ∈ E(G) there are metrics d 1 , d 2 ∈ M(w) such that if n = i (mod 2m) and i = 1, . . ., 2m. Now, using the Kuratowski embedding, we will define a metric space (X, d) as a subset of the k-dimensional normed vector space l ∞ k with k = |V (G)| and the norm For every n ∈ N, the Kuratowski embedding can be defined as: and consider X with the metric d induced by the norm · ∞ . We claim that G X,r (ρ X , ν 0 ) and G(w, r) are isomorphic as weighted rooted graphs. The next part of the proof is similar to the corresponding reasoning from Example 4.14 in [1].
It follows directly from (4.36) that  By (4.37), for n ∈ N, there are j ∈ N and v = v(n) ∈ V (G) satisfying the equality x n = K j (v). It is well known that the Kuratowski embeddings are distance preserving (see, for example, [3, the proof of Theorem III.8.1]). Consequently, we have Now (4.39) implies v = r for all sufficiently large n. Moreover, using (4.34) and (4.40) we obtain n = j if n is large enough. Hence, ifx = (x n ) n∈N belongs to Seq(X,r) anddr(x) > 0, then for every sufficiently large n there is v(n) ∈ V (G) such that 1 r n x n ∞ = w 0 (v(n)).
Since the labeling w 0 : holds for all sufficiently large n, then we havex ∈ Seq(X,r). Thus there is a bijection f : V (G) → X(G X,r ) such that f (r) =X 0 ∞,r = ν 0 and, by (4.40), for every v ∈ V (G). It is easy to prove that f is an isomorphism of G(w) and G X,r (ρ X ). Indeed, if u and v are distinct vertices of G and then, using (4.34), we obtain for all sufficiently large n ∈ N, where i ∈ {1, . . . , 2m} and i = n (mod 2m Thus G(w, r) and G X,r (ρ X , ν 0 ) are isomorphic as weighted rooted graphs.
The following corollary of Theorem 4.9 gives us a solution of Problem 2.4 for the case of finite graphs. Proof. If |V (G)| = 1, then it follows from Example 1.9. Now let r be a dominating vertex of G and let |V (G)| ≥ 2 hold. Define a weight w such that 1 < w(e) < 2, for all e ∈ E(G), and w(e 1 ) = w(e 2 ) if e 1 = e 2 . Then conditions (i)-(iii) of Theorem 4.9 are satisfied, and, consequently, there exist (X, d) andr such that G(w, r) and G X,r (ρ X , ν 0 ) are isomorphic as weighted rooted graphs. Thus, G(r) and G X,r (ν 0 ) are isomorphic as rooted graphs.
The converse statement follows directly form Proposition 2.2.
Corollary 4.11. Let (Y, δ) be a finite nonempty metric space. Then the following statements are equivalent.
(i) There is y * ∈ Y such that δ(y * , x) = δ(y * , z) (4.42) holds whenever x and z are distinct points of Y .
(ii) There are an unbounded metric space (X, d) and a scaling sequencer such that (X, d) has the unique pretangent space at infinity with respect tor and this pretangent space is isometric to (Y, δ).
Proof. (i) ⇒ (ii) Suppose (i) holds. To prove (ii) it suffices to consider a finite, weighted rooted graph G = G(w, r) such that: • G is complete, i.e., {x, y} ∈ E(G), whenever x and y are distinct points of Y ; • The equality w({x, y}) = δ(x, y) holds for every {x, y} ∈ E(G); • The root r coincides with a point y * for which (4.42) holds for all distinct x, y ∈ Y . Theorem 4.9 implies the existence of (X, d) andr having the desirable properties.
It is known that the maximum number f (n) of maximal cliques possible in a finite graph with n ≥ 2 vertices satisfies the equality where ⌊·⌋ is the floor function. (See [9] and [16] for the proof and related results.) Corollary 4.12. Let (X, d) be an unbounded metric space and letr be a scaling sequence. Then, we have either or Ω X ∞,r ≥ c if |V (G X,r )| is infinite, where Ω X ∞,r is the cardinal number of distinct pretangent spaces to (X, d) at infinity with respect tor and f satisfies equality (4.43).
Proof. If the labeling ρ 0 : V (G X,r ) → R + is not injective, then by Lemma 3.5 there is an independent set I ⊆ V (G X,r ) such that |I| = c.
For every ν ∈ I there is Ω X ∞,r such that ν ∈ Ω X ∞,r and by virtue the fact that I is independent, the distinct points of I belong to distinct pretangent spaces. Hence Ω X ∞,r ≥ c holds. Let ρ 0 : V (G X,r ) → R + be injective. If |V (G X,r )| ≤ 2, then statement (iii) of Proposition 1.5 and Definition 1.4 imply Ω X ∞,r = 1. Assume now that 3 ≤ |V (G X,r )| < ∞. The point ν 0 =X 0 ∞,r is a dominating vertex of G X,r . Consequently, there is an one-to-one correspondence between the maximal cliques of G X,r and the maximal cliques of the vertex-deleted subgraph G X,r − ν 0 . Since 2 ≤ |V (G X,r − ν 0 )| < ∞, we may use function (4.43) to obtain the desirable estimation. Remark 4.13. Inequality (4.44) is the best possible in the sense that, for every n ∈ N, there exist an unbounded metric space (X, d) and a scaling sequencer such that |V (G X,r )| = n and Ω X ∞,r = 1 if |V (G X,r )| ≤ 2 f (|V (G X,r )| − 1) if 3 ≤ |V (G X,r )| < ∞.
It directly follows from Corollary 4.10 that the vertex-deleted subgraph (G X,r − ν 0 ) of the graph G X,r can be isomorphic to arbitrary finite graph G with |V (G)| = |V (G X,r )| − 1.
We conclude this section by a brief discussion of conditions (ii) and (iii) of Theorem 4.9.
By Lemma 4.1 condition (ii) means that every weighted cycle C ⊆ G(w) is metrizable with the weight induced from G(w). Furthermore, it was shown that condition (iii) is equivalent to the fact that the vertex set V (C) of every uniquely metrizable cycle C ⊆ G(w) is a clique in G(w).
For an arbitrary metrizable cycle C = C(w) there is a circle S in the plane and a finite subset A of S such that |V (C)| = |A| and, for every {u, v} ∈ E(C), there are a, b ∈ A for which the length of the minor arc between a and b equals to w({u, v}). So we can consider a set A together with the metric defined by the minor arc length as a result of metrization of the weighted cycle C(w). We know that this metrization is unique (up to an isometry) if and only if  The unique metrization of a weighted cycle C(w) satisfying equality (4.45) can also be represented as a finite set of points on the real line with the standard metric d(x, y) = |x − y| (see Figure 3). The last representation is closely connected to the important concept of "metric betweenness" which was introduces by Menger [15] in the following form.
Let (X, d) be a metric space and let x, y and z be different points of X. One says that y lies between x and z if d(x, z) = d(x, y) + d(y, z).
It is easy to verify that, for three different points x, y, z ∈ X, we have 2 max{d(x, y), d(x, z), d(y, z)} = d(x, y) + d(x, z) + d(y, z) if and only if one of these points lies between the other two points. Thus equality (4.45) can be considered as a generalization of the "metric betweenness" relation to the case of weighted graphs. Characteristic properties of ternary relations that are "metric betweenness" relations were determined by Wald in [22]. Later, the problem of metrization of "betweenness" relations (not necessarily by real-valued metrics) was considered in [14,17,21]. Analogs of the classical Sylvester-Gallai and Bruijn-Erdös theorems for "metric betweenness" relations have recently been obtained in [5][6][7].