The Rado graph and the Urysohn space Peter J. Cameron p.j.cameron@qmul.ac.uk Reading Combinatorics Conference, 18 May 2006 Rado’s graph Consider countable graphs following condition ( ∗ ) : In 1964, Rado constructed a universal graph as follows: The vertex set is the set of natural num- Given any two finite disjoint sets U and bers (including zero). V of vertices, there is a vertex z joined to For i , j ∈ N , i < j , then i and j are joined if and every vertex in U and to no vertex in V . only if the i th digit in j (in base 2) is 1. Clearly a graph satisfying ( ∗ ) is universal. A Another construction: “back-and-forth” argument shows that any two Let P 1 denote the set of primes congruent to 1 countable graphs satisfying ( ∗ ) are isomorphic, mod 4. According to the Quadratic Reciprocity and a small modification shows that any such Law, for p , q ∈ P 1 , p is a square mod q if and only graph is homogeneous. if q is a square mod p . Join p to q if this holds. Thus, Rado’s graph is the unique countable This graph is isomorphic to Rado’s. graph (up to isomorphism) satisfying condition ( ∗ ) . Universality and homogeneity Rado showed that R is universal : every finite or Measure and category countable graph can be embedded in R . There are two natural ways of saying that a set It is also true (though not really obvious) that R of countable graphs is “large”. is homogeneous : every isomorphism between finite Choose a fixed countable vertex set, and enu- subgraphs of R extends to an automorphism of R . merate the pairs of vertices: { x 0 , y 0 } , { x 1 , y 1 } , . . . There is a probability measure on the set of Exercise: Find an automorphism interchanging graphs, obtained by choosing independently with 0 and 1. probability 1/2 whether x i and y i are joined, for all i . Now a set of graphs is “large” if it has probabil- Uniqueness ity 1. Rado’s graph is the unique (up to isomorphism) There is a complete metric on the set of graphs: the distance between two graphs is 1/2 n if n is graph which is countable, universal and homoge- neous. minimal such that x n and y n are joined in one In fact, it suffices for this statement to assume graph but not the other. Now a set of graphs is universality for finite graphs (that is, every finite “large” if it is residual in the sense of Baire cate- graph can be embedded as an induced subgraph) gory, that is, contains a countable intersection of and homogeneity. open dense sets. Recognition Ubiquity 1
It is now quite easy to show that the set of count- Proof. Enumerate the edges of R : e 1 , e 2 , . . .. Sup- pose we have found disjoint subgraphs G ′ 1 , . . . , G ′ able graphs satisfying ( ∗ ) (that is, the set of graphs n isomorphic to R ) is “large” in both the senses just isomorphic to G 1 , . . . , G n and containing e 1 , . . . , e n . Then R \ ( G ′ 1 ∪ · · · ∪ G ′ n ) is isomorphic to R , so described. contains a spanning subgraph G ′ In fact, condition ( ∗ ) with fixed sets U and V n + 1 isomorphic to is satisfied in an open dense set of graphs with G n + 1 ; moreover, since the automorphism group of full measure, and there are only countably many R is edge-transitive, we may assume that this sub- choices of the pair ( U , V ) . graph contains e n + 1 , if this edge is not already cov- ered by G ′ 1 , . . . , G ′ Thus, Rado’s graph is the countable random graph , n . as well as the generic countable graph . Automorphisms Indestructibility The automorphism group of R is a very interest- A number of operations can be applied to R ing group. Some of its properties: without changing its isomorphism type. These in- clude • Aut ( R ) has cardinality 2 ℵ 0 ; • deleting any finite set of vertices; • Aut ( R ) is simple; • adding or deleting any finite set of edges; • Aut ( R ) has the small index property , that is, any subgroup of index less than 2 ℵ 0 contains • more generally, adding or deleting any set of edges such that only finitely many are inci- the pointwise stabiliser of a finite set of ver- dent with each vertex; tices; • taking the complement. • Aut ( R ) contains a generic conjugacy class , one that is residual in the whole group; Pigeonhole property • Aut ( R ) contains a copy of every finite or A countable graph G is said to have the pigeon- countable group. hole property if, whenever the vertex set of G is par- titioned into two parts in any manner, the induced subgraph on one of these parts is isomorphic to G . Homomorphisms Rado’s graph has the pigeonhole property. A homomorphism of a graph G is a map from G Indeed, there are just three countable graphs to G which maps edges to edges. The endomor- with the pigeonhole property: the complete graph, phisms of any graph G (the homomorphisms from the null graph, and Rado’s graph. G to G ) form a monoid (a semigroup with identity). The endomorphism monoid of R contains a Spanning subgraphs copy of every finite or countable monoid. A countable graph G is a spanning subgraph of R if and only if, for any finite set W of vertices of Homomorphism-homogeneity G , there is a vertex Z joined to no vertex in W . Recall that a graph G is homogeneous if every In particular, any locally finite graph is a span- isomorphism between finite subgraphs of G can be ning subgraph of R . extended to an isomorphism from G to G . Dually, R is a spanning subgraph of G if and only if any finite set of vertices of G have a com- We obtain new classes of graphs by replac- mon neighbour. ing “isomorphism” by “homomorphism” (or “monomorphism”) in this definition. What is known? Factorisations Theorem 1. Let G 1 , G 2 , . . . be a sequence of locally fi- • Every graph containing R as a span- nite countable non-null graphs. Then R can be parti- ning subgraph is homomorphism- and tioned into subgraphs isomorphic to G 1 , G 2 , . . . . monomorphism-homogeneous. 2
for i , j = 1, . . . , n . • If a countable graph G has the property that every monomorphism between finite sub- Thus the possible distances are chosen from a cone in R n . graphs extends to a homomorphism of G , then either G contains R as a spanning sub- graph, or there is a bound on the size of claws Ubiquity K 1, n in G . Thus we have both a measure and a metric on the set of countable metric spaces. For the mea- Apart from disjoint unions of complete graphs sure, use any natural probability measure on the (which contain no K 1,2 ), no homomorphism- cone in R n at each step, for example, the restric- homogeneous graphs of bounded claw size are tion of a Gaussian measure on the whole space. known. Anatoly Vershik showed that Polish spaces • the completion of a random countable metric There is a complete metric space with properties space is isometric to U with probability 1; remarkably similar to those of Rado’s graph. • the set of countable metric spaces whose com- A complete metric space will not usually be pletion is U is residual in the set of all count- countable. Instead we require it to be separable , able metric spaces. that is, to have a countable dense subset. A Polish space is a complete separable metric In other words, Urysohn space is the random Pol- space. ish space , and the generic Polish space . Thus, the completion of any countable metric Unfortunately we don’t have a simple direct space is a Polish space. (This is analogous to the construction of U . construction of R from Q .) Rado and Urysohn Urysohn space Any countable dense subset of U carries the In a posthumous paper published in 1927, structure of Rado’s graph R (in many different P. S. Urysohn showed: ways). Simply partition the set of distances which occur into two subsets E and N (satisfying some Theorem 2. There is a unique Polish space which is weak restrictions), and join x to y if d ( x , y ) ∈ E . • universal, that is, every Polish space can be iso- Hence, if a group G acts as an isometry group of metrically embedded into it; U with a countable dense orbit, then G acts as an automorphism group of R . • homogeneous, that is, every isometry between fi- nite subsets can be extended to an isometry of the Examples whole space. The Urysohn space admits an isometry all of whose orbits are dense. So the infinite cyclic group We denote Urysohn space by U . is an example of a group acting on R . (In fact, if we choose a “random countable circulant graph”, it is Constructing a Polish space isomorphic to R with probability 1. To construct a Polish space, build a countable The countable elementary abelian 2-group also metric space one point at a time and take its com- acts on U with dense orbits. pletion. The reverse implication is false. The countable Suppose that points a 1 , . . . , a n have been con- elementary abelian 3-group acts on R but not on structed and their distances d ( a i , a j ) specified. We U . want to add a new point a n + 1 with distances d ( a n + 1 , a i ) = x i for i = 1, . . . , n . These distances Ramsey theory must satisfy x i ≥ 0 for i = 1, . . . , n and There is a close connection between homogene- | x i − x j | ≤ d ( a i , a j ) ≤ x i + x j ity and Ramsey theory. 3
Hubicka and Neˇ setˇ ril have shown that, if a countably infinite structure carries a total order and the class of its finite substructures is a Ramsey class, then the infinite structure is homogeneous. The finite substructures of R are the finite graphs, which do form a Ramsey class. The converse is false in general, but Neˇ setˇ ril re- cently showed that the class of finite metric spaces is a Ramsey class. 4
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