The Rado graph and the Urysohn space Peter J. Cameron [email protected] 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- bers (including zero). Given any two finite disjoint sets U and 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 ith digit in j (in base 2) is 1. (∗) Another construction: Clearly a graph satisfying is universal. A Let P denote the set of primes congruent to 1 “back-and-forth” argument shows that any two 1 (∗) mod 4. According to the Quadratic Reciprocity countable graphs satisfying are isomorphic, and a small modification shows that any such Law, for p, q ∈ P1, p is a square mod q if and only if q is a square mod p. Join p to q if this holds. graph is homogeneous. 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: {x0, y0}, {x1, y1},... Exercise: Find an automorphism interchanging There is a probability measure on the set of 0 and 1. graphs, obtained by choosing independently with probability 1/2 whether xi and yi 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: graph which is countable, universal and homoge- the distance between two graphs is 1/2n if n is neous. minimal such that xn and yn 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: e1, e2, . .. Sup- 0 0 able graphs satisfying (∗) (that is, the set of graphs pose we have found disjoint subgraphs G1,..., Gn isomorphic to R) is “large” in both the senses just isomorphic to G1,..., Gn and containing e1,..., en. 0 0 described. Then R \ (G1 ∪ · · · ∪ Gn) is isomorphic to R, so 0 In fact, condition (∗) with fixed sets U and V contains a spanning subgraph Gn+1 isomorphic to is satisfied in an open dense set of graphs with Gn+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 en+1, if this edge is not already cov- 0 0 Thus, Rado’s graph is the countable random graph, ered by G1,..., Gn. 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 2ℵ0 ; • 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, • more generally, adding or deleting any set of any subgroup of index less than 2ℵ0 contains 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 A countable graph G is said to have the pigeon- • Aut(R) contains a copy of every finite or hole property if, whenever the vertex set of G is par- countable group. 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 G, there is a vertex Z joined to no vertex in W. Homomorphism-homogeneity In particular, any locally finite graph is a span- Recall that a graph G is homogeneous if every ning subgraph of R. isomorphism between finite subgraphs of G can be Dually, R is a spanning subgraph of G if and extended to an isomorphism from G to G. 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. Factorisations What is known? Theorem 1. Let G1, G2,... 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 G1, G2,.... monomorphism-homogeneous. 2 • If a countable graph G has the property that for i, j = 1, . , n. every monomorphism between finite sub- Thus the possible distances are chosen from a graphs extends to a homomorphism of G, cone in Rn. then either G contains R as a spanning sub- graph, or there is a bound on the size of claws Ubiquity K in G. 1,n Thus we have both a measure and a metric on Apart from disjoint unions of complete graphs the set of countable metric spaces. For the mea- (which contain no K ), no homomorphism- sure, use any natural probability measure on the 1,2 n homogeneous graphs of bounded claw size are cone in R at each step, for example, the restric- known. tion of a Gaussian measure on the whole space. 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. A complete metric space will not usually be • the set of countable metric spaces whose com- countable. Instead we require it to be separable, pletion is U is residual in the set of all count- that is, to have a countable dense subset. able metric spaces. 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 Theorem 2. There is a unique Polish space which is occur into two subsets E and N (satisfying some 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 whole space. Examples The Urysohn space admits an isometry all of We denote Urysohn space by U. whose orbits are dense. So the infinite cyclic group 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 ,..., a have been con- 1 n elementary abelian 3-group acts on R but not on structed and their distances d(a , a ) specified. We i j U. want to add a new point an+1 with distances d(an+1, ai) = xi for i = 1, . , n. These distances must satisfy xi ≥ 0 for i = 1, . , n and Ramsey theory There is a close connection between homogene- |xi − xj| ≤ d(ai, aj) ≤ xi + xj ity and Ramsey theory.