The Rado Simplicial Complex

A Rado simplicial complex X is a generalisation of the well-known Rado graph. X is a countable simplicial complex which contains any countable simplicial complex as its induced subcomplex. The Rado simplicial complex is highly symmetric, it is homogeneous: any isomorphism between finite induced subcomplexes can be extended to an isomorphism of the whole complex. We show that the Rado complex X is unique up to isomorphism and suggest several explicit constructions. We also show that a random simplicial complex on countably many vertices is a Rado complex with probability 1. The geometric realisation |X| of a Rado complex is contractible and is homeomorphic to an infinite dimensional simplex. We also prove several other interesting properties of the Rado complex X, for example we show that removing any finite set of simplexes of X gives a complex isomorphic to X.


Introduction
The Rado graph Γ is a remarkable combinatorial object, it can be characterised by its universality and homogeneity. The graph Γ is universal in the sense that any graph with finitely or countably many vertices is isomorphic to an induced subgraph of Γ. Besides, any isomorphism of between finite induced subgraphs of Γ can be extended to the whole Γ (homogeneity). Γ is unique up to isomorphism and there are many different ways of constructing it explicitly. Erds and Rnyi [5] showed that an infinite random graph is isomorphic to Γ with probability 1. The Rado graph Γ was introduced by Richard Rado [10] in 1964, see also previous construction of W. Ackermann [1]. One may mention surprising robustness of Γ: removing any finite set of vertices and edges produces a graph isomorphic to Γ. Many other interesting results about the Rado graph Γ are known, we refer the reader to the survey [3].
The goal of the present paper is to describe a Rado simplicial complex X. It is a simplicial complex with countably many vertices which is universal and homogeneous; we show that these two properties characterise X up to isomorphism. R. Rado mentions universal simplicial complexes in his paper [10] and gives a specific construction. The 1-dimensional skeleton of X is a Rado graph. We show that X can also be characterised by its ampleness which is a high-dimensional generalisation of the well-known extension property of the Rado graph. We also prove that X is robust, i.e. removing any set of finitely many simplexes produces a simplicial complex isomorphic to X. If the set of vertices of X is partitioned into finitely many parts, the simplicial complex induced on at least one of these parts is isomorphic to X. The link of any simplex of X is a Rado complex. We prove that a random infinite simplcial complex is isomorphic to X with probability 1. Besides, we show that the geometric realisation |X| of the Rado complex is homeomorphic to the infinite dimensional simplex.
2. The Definition of the Rado complex 2.1. Basic terminology. A simplicial complex X is a set of vertexes V (X) and a set of non-empty finite subsets of V (X), called simplexes such that any vertex v ∈ V (X) is a simplex {v} and any subset of a simplex is a simplex. A simplicial complex X is said to be countable (finite) if its vertex set V (X) is countable (finite). The symbol F (X) stands for the set of all simplexes of X. For a simplex σ ∈ F (X) we shall also write σ ⊂ X.
Two simplicial complexes are isomorphic if there is a bijection between their vertex sets which induces a bijection between the sets of simplexes.
The standard simplex ∆ n has the set of vertexes {1, 2, . . . , n} with all non-empty subsets as simplexes. Another standard simplex is ∆ N ; its vertex set is N = {1, 2, . . . } and all non-empty finite subsets of N are simplexes.
A simplicial subcomplex Y ⊂ X is said to be induced if any simplex σ ∈ F (X) with all its faces in V (Y ) belongs to F (Y ). The induced subcomplex Y ⊂ X is completely determined by the set of its vertices, V (Y ) ⊂ V (X). We shall use the notation For a vertex v ∈ V (X) the symbol Lk X (v) stands for the link of v; the latter is the union of simplexes σ of X with v / ∈ σ and vσ ⊂ X.
(1) A countable simplicial complex X is said to be universal if any countable simplicial complex is isomorphic to an induced subcomplex of X.
(2) We say that X is homogeneous if for any two finite induced subcomplexes X U , X U ′ ⊂ X and for any isomorphism f : A countable simplicial complex X is a Rado complex if it is universal and homogeneous.
It is clear that the 1-skeleton of a Rado complex is a Rado graph; the latter can be defined as a universal and homogeneous graph having countably many vertexes, see [3].
We prove in this paper: Theorem 2. Rado simplicial complexes exist and any two Rado complexes are isomorphic.
The following property is a useful criterion of being a Rado complex: Definition 3. We shall say that a countable simplicial complex X is ample if for any finite subset U ⊂ V (X) and for any simplicial subcomplex Remark 4. Condition (1) can equivalently be expressed as where U ′ = U ∪ {v} and vA denotes the cone with apex v and base A. In literature the cone vA is also sometimes denoted v * A, the simplicial join of a vertex v and complex A.

Remark 5.
Suppose that X is a simplicial complex with countable set of vertexes V (X). One may naturally consider exhaustions U 0 ⊂ U 1 ⊂ U 2 ⊂ · · · ⊂ V (X) consisting of finite subsets U n satisfying ∪U n = V (X). In order to check that X is ample as defined in Definition 3 it is sufficient to verify that for any n ≥ 0 and for any subcomplex Remark 6. Suppose that X is an ample simplicial complex. Given finitely many distinct vertexes u 1 , . . . , u m , v 1 , . . . , v n ∈ V (X), there exists a vertex z ∈ V (X) which is adjacent to u 1 , . . . , u m and nonadjacent to v 1 , . . . , v n . To see this we apply Definition 3 with U = {u 1 , . . . , u m , v 1 , . . . , v n } and A = {u 1 , . . . , u m }. This shows that the 1-skeleton of a Rado complex satisfies the defining property of the Rado graph [3]. This also shows that ampleness is a high dimensional generalizaton of this graph property.
The following property of ample complexes will be useful in the sequel. Lemma 7. Let X be an ample complex and let L ′ ⊂ L be a pair consisting of a finite simplical complex L and an induced subcomplex L ′ . Let f ′ : L ′ → X U ′ be an isomorphism of simplicial complexes, where U ′ ⊂ V (X) is a finite subset. Then there exists a finite subset U ⊂ V (X) containing U ′ and an isomorphism f : Proof. It is enough to prove this statement under an additional assumption that L has a single extra vertex, i.e. V (L) − V (L ′ ) = 1. In this case L is obtained from L ′ by attaching a cone wA where w ∈ V (L) − V (L ′ ) denotes the new vertex and A ⊂ L ′ is a subcomplex (the base of the cone). Applying the defining property of the ample complex to the subset Proof. Suppose X is a Rado complex, i.e. X is universal and homogeneous. Let U ⊂ V (X) be a finite subset and let A ⊂ X U be a subcomplex of the induced complex. Consider an abstract simplicial complex L = X U ∪ wA which obtained from X U by adding a cone wA with vertex w and base A where X U ∩ wA = A. Clearly, V (L) = U ∪ {w}. By universality, we may find a subset U ′ ⊂ V (X) and an isomorphism g : L → X U ′ . Denoting w 1 = g(w), A 1 = g(A) and U 1 = g(U ) we have X U ′ = X U 1 ∪ w 1 A 1 . Obviously, g restricts to an isomorphism g|X U : X U → X U 1 . By the homogeneity property we can find an isomorphism F : X → X with F |X U = g|X U . Denoting v = F −1 (w 1 ) we shall have X U ∪{v} = X U ∪ vA as required, see Remark 4. Now suppose that X is ample. To show that it is universal consider a simplicial complex L with at most countable set of vertexes V (L). We may find a chain of induced subcomplexes L 1 ⊂ L 2 ⊂ . . . with ∪L n = L and each complex L n has exactly n vertexes. Then L n+1 obtained from L n by adding a cone v n+1 A n where v n+1 is the new vertex and A n ⊂ L n is a simplicial subcomplex. We argue by induction that we can find a chain of subsets U 1 ⊂ U 2 ⊂ · · · ⊂ V (X) and isomorphisms f n : L n → X Un satisfying f n+1 |L n = f n . If U n and f n are already found then the next set U n+1 and the isomorphism f n+1 exist because X is ample: we apply Definition 3 with U = U n and A = f n (A n ) and we set U n+1 = U n ∪ {v} where v is the vertex given by Definition 3. The sequence of maps f n defines an injective map f : V (L) → V (X) and produces an isomorphism between L and the induced subcomplex X f (V (L)) .
The fact that any ample complex is homogeneous follows from Lemma 9 below. We state it in a slightly more general form so that it also implies the uniqueness of Rado complexes. Lemma 9. Let X and X ′ be two ample complexes and let L ⊂ X and L ′ ⊂ X ′ be two induced finite subcomplexes. Then any isomorphism f : L → L ′ can be extended to an isomorphism F : X → X ′ .
Proof. We shall construct chains of subsets of the sets of vertexes U 0 ⊂ U 1 ⊂ · · · ⊂ V (X) and We shall also construct isomorphisms f n : X Un → X U ′ n satisfying f 0 = f and f n+1 |X Un = f n . The whole collection {f n } will then define a required isomorphism F : To constructs these objects we shall use the well known back-and-forth procedure. Enu- We act by induction and describe U n , U ′ n and f n assuming that the objects U i , U ′ i and f i : The procedure will depend on the parity of n. For n odd we find the smallest j with v j / ∈ U n−1 and set U n = U n−1 ∪ {v j }. Applying Lemma 7 to the simplicial complexes L = X Un , L ′ = X U n−1 and the isomorphism f n−1 : For n even we proceed in the reverse direction. We find the smallest Corollary 10. Any two Rado complexes are isomorphic.
Proof. This follows from Theorem 8 with subsequent applying Lemma 9 with L = L ′ = ∅.
Remark 11. In Definition 1 we defined universality with respect to arbitrary countable simplicial subcomplexes. A potentially more restrictive definition dealing only with finite subcomplexes together with homogeneity is in fact equivalent to Definition 1; this follows from the arguments used in the proof of Theorem 8.

Remark 12.
There is a variation of the whole study presented in this paper when one is interested in simplicial complexes containing no induced subcomplexes isomorphic to ∂∆ d , for a fixed d > 0. Note that ∂∆ d is the simplicial complex with the vertex set {1, 2, . . . , d} and with all non-empty proper subsets of {1, 2, . . . , d} as simplexes. Simplicial complexes with this property can also be characterised as having no external d-dimensional simplexes.

Definition 3 can be modified for this case:
It can be shown (similarly to the above) that d-ample simplicial complexes exist and are universal and homogeneous with respect to the class of complexes with no induced ∂∆ d .
Moreover Theorem 27 can also be generalised: random infinite simplicial complexes are universal with probability one for this specific class (see §6 and §7) when the probability parameters p σ satisfy p σ = 1 for all simplexes σ with dim σ = d and 0 < p − ≤ p σ ≤ p + < 1 for dim σ = d.
1. An inductive construction. One may construct a Rado simplicial complex X inductively as the union of a chain of finite induced simplicial subcomplexes Here X 0 is a single point and each complex X n+1 is obtained from X n by first adding a finite set of vertices v(A), labeled by subcomplexes A ⊂ X n (including the case when A = ∅); secondly, we construct the cone v(A) * A with apex v(A) and base A, and thirdly we attach each such cone v(A) * A to X n along the base A ⊂ X n . Thus, To show that the complex X = ∪ n≥0 X n is ample, i.e. a Rado complex, we refer to Remark 5 and observe that any subcomplex 3.2. An explicit construction. Here we shall give an explicit construction of a Rado complex X. To describe it we shall use the sequence {p 1 , p 2 , . . . , } of all primes in increasing order, where p 1 = 2, p 2 = 3, etc.
The set of vertexes V (X) is the set of all positive integers N. Each simplex of X is uniquely represented by an increasing sequence a 0 < a 1 < · · · < a k with certain properties. Subsequences of a 0 < a 1 < · · · < a k are obtained by erasing one or more elements in the sequence.
We shall say that an increasing sequence of positive integers 0 < a 0 < a 1 < · · · < a k represents a simplex of X if all its proper subsequences are in X and additionally the p a 0 p a 1 . . . p a k−1 -th binary digit of a k is 1.
Proposition 14. The obtained simplicial complex X is Rado.
Proof. With any increasing sequence σ of positive integers 0 < a 0 < a 1 < · · · < a k we associate the product N σ = p a 0 p a 1 . . . p a k , which is an integer without multiple prime factors. Note that for two such increasing sequences σ and σ ′ one has N σ = N σ ′ if and only if σ is identical to σ ′ .
Given a finite subset U ⊂ V (X) and a simplicial subcomplex where The binary expansion of v has ones exactly on positions N σ where σ ∈ F (A) and it has zeros on all other positions except an additional 1 at position K U . Note that K U > N σ for any simplex σ ⊂ X U . In particular, we see that vertex v defined by (4) satisfies v > w for any w ∈ U .
Consider a simplex σ ⊂ X U . By definition, the simplex vσ with apex v and base σ lies in X if and only if the N σ -th binary digit of v is 1. We see from (4) that it happens if and only if σ ⊂ A. This means that Lk X (v) ∩ X U = A and hence the complex X is a Rado complex.

Some properties of the Rado complex
Lemma 15. Let X be a Rado complex, let U ⊂ V (X) be a finite set and let A ⊂ X U be a subcomplex. Let Z U,A ⊂ V (X) denote the set of vertexes v ∈ V (X) − U satisfying (1). Then Z U,A is infinite and the induced complex on Z U,A is also a Rado complex.
Proof. Consider a finite set {v 1 , . . . , v N } ⊂ Z U,A of such vertexes. One may apply Definition 3 to the set U 1 = U ∪ {v 1 , . . . , v N } and to the subcomplex A ⊂ X U 1 to find another vertex v N +1 satisfying the condition of Definition 3. This shows that Z U,A must be infinite.
Corollary 16. Let X be a Rado complex and let Y be obtained from from X by selecting a finite number of simplexes F ⊂ F (X) and deleting all simplexes σ ∈ F (X) which contain simplexes from F as their faces. Then Y is also a Rado complex.
Proof. Let U ⊂ V (Y ) be a finite subset and let A ⊂ Y U be a subcomplex. We may also view U as a subset of V (X) and then A becomes a subcomplex of X U since Y U ⊂ X U . The set of vertexes v ∈ V (X) satisfying Lk X (v) ∩ X U = A is infinite (by Lemma 15) and thus we may find a vertex v ∈ V (X) which is not incident to simplexes from the family F .
Corollary 17. Let X be a Rado complex. If the vertex set V (X) is partitioned into a finite number of parts then the induced subcomplex on at least one of these parts is a Rado complex.
Proof. It is enough to prove the statement for partitions into two parts. Let V (X) = V 1 ⊔V 2 be a partition; denote by X 1 and X 2 the subcomplexes induced by X on V 1 and V 2 correspondingly. Suppose that none of the subcomplexes X 1 and X 2 is Rado. Then for each i = 1, 2 there exists a finite subset Since X is Rado we may find a vertex v ∈ V (X) with Lk X ∩X U = A. Then v lies in V 1 or V 2 and we obtain a contradiction, since Lk Lemma 18. In a Rado complex X, the the link of every simplex is a Rado complex.
Proof. Let Y = Lk X (σ) be the link of a simplex σ ∈ X. To show that Y is Rado, let U ⊂ V (Y ) be a subset and let A ⊂ Y U be a subcomplex. We may apply the defining property of the Rado complex to the subset U ′ = U ∪ V (σ) ⊂ V (X) and to the subcomplex A ⊔σ ⊂ X U ′ ; hereσ denotes the subcomplex containing the simplex σ and all its faces. We obtain a vertex w ∈ V (X) − U ′ with Lk X (w) ∩ X U ′ = A ⊔σ or equivalently, X U ′ ∪w = X U ′ ∪ wA, see Remark 4. Note that w ∈ Y = Lk X (σ) since the simplex wσ is in X. Besides, Y U ∪w = Y U ∪ wA. Hence we see that the link Y is also a Rado complex.

Geometric realisation of the Rado complex
Recall that for a simplicial complex X the geometric realisation |X| is the set of all functions α : V (X) → [0, 1] such that the support supp(α) = {v; α(v) = 0} is a simplex of X (and hence finite) and v∈X α(v) = 1, see [12]. For a simplex σ ∈ F (X) the symbol |σ| denotes the set of all α ∈ |X| with supp(α) ⊂ σ. The set |σ| has natural topology and is homeomorphic to the linear simplex lying in an Euclidean space. The weak topology on the geometric realisation |X| has as open sets the subsets U ⊂ |X| such that U ∩ |σ| is open in |σ| for any simplex σ.
Lemma 19. Let X be a Rado complex. Then there exists a sequence of finite subsets U 0 ⊂ U 1 ⊂ U 2 ⊂ · · · ⊂ V (X) such that ∪U n = V (X) and for any n = 0, 1, 2, . . . the induced simplicial complex X Un is isomorphic to a triangulation L n of the standard simplex ∆ n+1 of dimension n. Moreover, for any n the complex L n is naturally an induced subcomplex of L n+1 and the isomorphisms f n : X Un → L n satisfy f n+1 |X Un = f n .
Proof. Let V (X) = {v 0 , v 1 , . . . } be a labelling of the vertices of X. One constructs the subsets U n and complexes L n by induction stating from U 0 = {v 0 } and L 0 = {v 0 }. Suppose that the sets U i and complexes L i with i ≤ n have been constructed. Consider the subset Clearly, the complex L ′ n+1 has the form X Un ∪ (v i * A) for some subcomplex A ⊂ X Un . We shall apply Lemma 7 to the abstract simplicial complexes L ′ n+1 and L n+1 , where L n+1 is a subdivision of the cone v i * X Un such that L ′ n+1 is an induced subcomplex of L n+1 . Lemma 7 gives a subset U n+1 ⊂ V (X) containing U ′ n+1 such that the induced complex X U n+1 is isomorphic to L n+1 . By construction L n+1 is a triangulation of v i * X Un , hence it is a triangulation of a simplex of dimension n + 1. Obviously, we have ∪U n = V (X).
Theorem 20. The Rado complex is isomorphic to a triangulation of the simplex ∆ N . In particular, the geometric realisation |X| of the Rado complex is homeomorphic to the infinite dimensional simplex |∆ N |.
Proof. It follows from the previous Lemma.
Note that the geometric realisation |X| of a Rado complex X (equipped with the weak topology) does not satisfy the first axiom of countability and hence is not metrizable. This follows from the fact that X is not locally finite. See [12], Theorem 3.2.8.
Corollary 21. The geometric realisation |X| of the Rado complex is contractible.
Proof. Corollary follows from the previous Theorem. We also give a short independent proof below. Let X be a Rado complex. By the Whitehead theorem we need to show that any continuous map f : S n → X is homotopic to the constant map. By the Simplicial Approximation theorem f is homotopic to a simplicial map g : S n → X. The image g(S n ) ⊂ X is a finite subcomplex. Applying the property of Definition 3 to the set of vertices U of g(S n ) and to the subcomplex A = X U we find a vertex v ∈ V (X) − U such that the cone vA is a subset of X. Since the cone is contractible, we obtain that g, which is equal the composition S n → A → vA → X, is null-homotopic.
Remark 22. The geometric realisation of a simplicial complex carries another natural topology, the metric topology, see [12]. The geometric realisation of X with the metric topology is denoted |X| d . While for finite simplicial complexes the spaces |X| and |X| d are homeomorphic, it is not true for infinite complexes in general. For the Rado complex X the spaces |X| and |X| d are not homeomorphic. Moreover, in general, the metric topology is not invariant under subdivisions, see [9], where this issue is discussed in detail. We do not know if for the Rado complex X the spaces |X| d and |∆ N | d are homeomorphic.

Infinite random simplicial complexes
We show in the following §7 that a random infinite simplicial complex is a Rado complex with probability 1, in a certain regime. In this section we prepare the grounds and describe the probability measure on the set of infinite simplicial complexes. 6.1. Let L be a finite simplicial complex. We denote by F (L) the set of simplexes of L; besides, V (L) will denote the set of vertexes of L. Suppose that with each simplex σ ⊂ L one has associated a probability parameter p σ ∈ [0, 1]. We shall use the notation q σ = 1 − p σ . Given a subcomplex A ⊂ L we may consider the set E(A|L) consisting of all simplexes of L which are not in A but such that all their proper faces are in A. Simplexes of E(A|L) are called external for A in L. As an example we mention that any vertex v ∈ L − A is an external simplex, v ∈ E(A|L).
With a subcomplex A ⊂ L one may associate the following real number For example, taking A = ∅ we obtain p(∅) = v∈V (L) q v , the product is taken with respect to all vertices v of L.
Lemma 23. One has A⊂L p(A) = 1, where A runs over all subcomplexes of L, including the empty subcomplex.
The proof can be found in §9.
6.2. Let ∆ = ∆ N denote the simplex spanned by the set N = {1, 2, . . . } of positive integers. We shall denote by Ω the set of all simplicial subcomplexes X ⊂ ∆. Each simplicial complex X ∈ Ω has finite or countable set of vertexes V (X) ⊂ N and any finite or countable simplicial complex is isomorphic to one of the complexes X ∈ Ω.
In other words, for X ∈ Ω the complex π n (X) ⊂ ∆ n is the subcomplex of X induced on the vertex set [n] ⊂ N.
For a subcomplex Y ⊂ ∆ n we shall consider the set Note that for n = n ′ the sets Z(Y, n) and Z(Y ′ , n ′ ) are either identical (if and only if Y = Y ′ ) of disjoint; for n > n ′ the intersection Z(Y, n) ∩ Z(Y ′ , n ′ ) is nonempty if and only if Y ∩ ∆ n ′ = Y ′ and in this case Z(Y, n) ⊂ Z(Y ′ , n ′ ). Note also that for n > n ′ and where Y j ⊂ ∆ n are all subcomplexes with Y j ∩ ∆ n ′ = Y ′ ; one of these subcomplexes Y j coincides with Y .
Let A denote the set of all subsets Z(Y, n) ⊂ Ω and ∅. The set A is a semi-ring, see [8], i.e. A is ∩-closed and for any A, B ∈ A the difference B − A is a finite union of mutually disjoint sets from A. We shall denote by A ′ the σ-algebra generated by A.
Example 24. Let U ⊂ N be a finite subset and let L be a simplicial complex with vertex set V (L) ⊂ U . Then the set {X ∈ Ω; X U = L} is the union of finitely many elements of the semi-ring A and in particular, {X ∈ Ω; X U = L} ∈ A ′ . Indeed, let n be an integer such that U ⊂ [n] and let Y j ⊂ ∆ n , for j ∈ I, be the list of all subcomplexes of ∆ n satisfying (Y j ) U = L; in other words, Y j induces L on U . Then the set {X ∈ Ω; X U = L} is the union ⊔ j∈I Z(Y j , n).
6.4. Next we defile a function µ : A → R as follows. Fix for every simplex σ ⊂ ∆ N a probability parameter p σ ∈ [0, 1]. The function will be called the system of probability parameters. Here σ runs over all simplexes σ ∈ F (∆ N ). We shall use the notation q σ = 1 − p σ .
For an integer n ≥ 0 and a subcomplex Y ⊂ ∆ n define Let us show that µ is additive. We know that the set Z(Y, n) equals the disjoint union Z(Y, n) = ⊔ j∈I Z(Y j , n + 1) (11) where Y j are all subcomplexes of ∆ n+1 satisfying Y j ∩ ∆ n = Y . One of these subcomplexes Y j 0 equals Y and the others contain the vertex (n + 1) and have the form In other words, all complexes Y j with j = j 0 are obtained from Y by adding a cone with apex n + 1 over a subcomplex A j ⊂ Y . Clearly, any subcomplex A j ⊂ Y may occur, including the empty subcomplex A j = ∅.
Note that Ω can be naturally viewed as the inverse limit of the finite sets Ω n , i.e. Ω = lim ← Ω n . Introducing the discrete topology on each Ω n we obtain the inverse limit topology on Ω and with this topology Ω is compact and totally disconnected; it is homeomorphic to the Cantor set. The sets Z(Y, n) ⊂ Ω are open and closed in this topology, hence they are compact.
Next we apply Theorem 1.53 from [8] to show that µ extends to a probability measure on the σ-algebra A ′ generated by A. This theorem requires for µ to be additive, σ-subadditive and σ-finite. By Theorem 1.36 from [8], σ-subadditivity is equivalent to σ-additivity. Recall that σ-additivity means that for A = ⊔ i A i (disjoint union of countably many elements of A) one has µ(A) = i µ(A i ). In our case, since the sets A i ⊂ Ω are open and closed and since Ω is compact, any representation A = ⊔ i A i must be finite and hence σ-additivity of µ follows from additivity.
For fixed n we have Ω = ⊔Z(Y, n) where Y runs over all subcomplexes of ∆ n (including ∅). Using additivity of µ and applying Lemma 23, we have µ(Ω) = Y ⊂∆n µ(Z(Y, n)) = 1. This shows that µ is σ-finite and hence by Theorem 1.53 from [8] µ extends to a probability measure on A ′ . The extended measure on A ′ will be denoted by the same symbol µ.
Example 25. As in Example 24, let U ⊂ N be a finite subset and let L be a simplicial complex with vertex set V (L) ⊂ U . Then Here ∆ U denotes the simplex spanned by U . The proof is left to the reader as an exercise.

Random simplicial complex in the medial regime is Rado
In this section we prove that an infinite random simplicial complex in the medial regime is a Rado complex with probability one. 1 Definition 26. We shall say that a system of probability parameters p σ , see (9), is in the medial regime if there exist 0 < p − < p + < 1 such that the probability parameter p σ satisfies p σ ∈ [p − , p + ] for any simplex σ ∈ F (∆ N ).
In other words, in the medial regime the probability parameters p σ are uniformly bounded away from zero and one.
Theorem 27. A random simplicial complex with countably many vertexes in the medial regime is a Rado complex, with probability one. 1 Finite simplicial complexes in the medial regime were studied in [7].
Proof. For a finite subset U ⊂ N and for a simplicial subcomplex A ⊂ ∆ U of the simplex ∆ U consider the set This set belongs to the σ-algebra A ′ and has positive measure, see Example 25.
Consider also the subset Ω U,L,A,v ⊂ Ω U,L consisting of all subcomplexes X ∈ Ω satisfying X U ∪v = L ∪ vA. Here A ⊂ L is a subcomplex and v ∈ N − U .
The conditional probability equals see (13). Note that the events Ω U,L,A,v , conditioned on Ω U,L for various v, are independent and the sum of their probabilities is ∞. Hence we may apply the Borel-Cantelli Lemma (see [8], page 51) to conclude that the set of complexes X ∈ Ω U,L such that X U ∪v = L ∪ vA for infinitely many vertices v has full measure in Ω U,L .
By taking a finite intersection with respect to all possible subcomplexes A ⊂ L this implies that the set Ω U,L * ⊂ Ω U,L of simplicial complexes X ∈ Ω U,L such that for any subcomplex A ⊂ L there exists infinitely many vertexes v with X U ∪v = L ∪ vA has full measure in Ω U,L .
Since Ω = ∩ U ∪ L⊂∆ U Ω U,L (where U ⊂ N runs over all finites subsets) we obtain that the set ∩ U ∪ L⊂∆ U Ω U,L * has measure 1 in Ω. But the latter set ∩ U ∪ L⊂∆ U Ω U,L * is exactly the set of all Rado simplicial complexes, see Lemma 15.

Random induced subcomplexes of a Rado complex
In this section we consider a different situation. Let X be a fixed Rado complex with vertex set V (X) = N. Suppose that each of the vertexes n ∈ N is selected at random with probability p n ∈ [0, 1] independently of the selection of all other vertexes. Denote by X ω the subcomplex of X induced on the selected set of vertexes. Here ω stands for the selection sequence, one may think that ω ∈ {0, 1} N . Under which condition on the sequence {p n } the complex X ω is Rado with probability 1?
Applying Borel-Cantelli Lemma we get: (1) If p n < ∞ then complex X ω has finitely many vertexes, with probability 1.
(3) If q n < ∞ (where q n = 1 − p n ) then the set of vertexes of X ω has a finite complement in N and hence X ω is a Rado complex with probability 1. In (3) we use Lemma 7.
The following result strengthens point (3) above: Lemma 28. Suppose that for some p > 0 one has p n ≥ p > 0 for any n ∈ N. Then X ω is a Rado complex with probability 1.
Proof. Denote by X n the subcomplex of X induced on the set [n] = {1, 2, . . . , n} ⊂ N. For a subcomplex A ⊂ X n consider the set of vertexes W (A, n) = {v ∈ N − [n]; Lk X (v) ∩ X n = A}.
We know that this set is infinite (see Lemma 15) and since each of the elements of this set is a vertex of X ω with probability at least p > 0 we obtain (using Borel -Cantelli) that with probability 1 the intersection V (X ω ) ∩ W (A, n) is infinite. Hence the set has measure 1 (as intersection of countably many sets of measure one). Here we use σadditivity of the Bernoulli measure. It is obvious that for any ω lying in the intersection (15) the induced complex X ω is ample and hence Rado.

Proof of Lemma 23
We obviously have Note that in the above sum, J can be also the empty set. Denote by A(J) ⊂ J the set of all simplexes σ ∈ J such that for any face τ ⊂ σ one has τ ∈ J. Note that A = A(J) is a simplicial complex, it is the largest simplicial subcomplex of L with F (A) ⊂ J. We also note that the set of external simplexes E(A|L) is disjoint from J.
We therefore see that the statement of Lemma 23 follows from (16).