The Rado graph, as numbered by Ackermann (1937) and Rado (1964).
In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with probability one) by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge. The names of this graph honor Richard Rado, Paul Erdős, and Alfréd Rényi, mathematicians who studied it in the early 1960s; it appears even earlier in the work of Wilhelm Ackermann (1937). The Rado graph can also be constructed non-randomly, by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of the natural numbers, or as an infinite Paley graph that has edges connecting pairs of prime numbers congruent to 1 mod 4 that are quadratic residues modulo each other.
Every finite or countably infinite graph is an induced subgraph of the Rado graph, and can be found as an induced subgraph by a greedy algorithm that builds up the subgraph one vertex at a time. The Rado graph is uniquely defined, among countable graphs, by an extension property that guarantees the correctness of this algorithm: no matter which vertices have already been chosen to form part of the induced subgraph, and no matter what pattern of adjacencies is needed to extend the subgraph by one more vertex, there will always exist another vertex with that pattern of adjacencies that the greedy algorithm can choose.
The Rado graph is highly symmetric: any isomorphism of its finite induced subgraphs can be extended to a symmetry of the whole graph.
The first-order logic sentences that are true of the Rado graph are also true of almost all random finite graphs, and the sentences that are false for the Rado graph are also false for almost all finite graphs. In model theory, the Rado graph is an example of the unique countable model of an ω-categorical theory.
In the mathematical field of graph theory, the Radograph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with...
Richard Rado FRS (28 April 1906 – 23 December 1989) was a German-born British mathematician whose research concerned combinatorics and graph theory. He...
constructed by Richard Rado and is now called the Radograph or random graph. More recent work has focused on universal graphs for a graph family F: that is...
only a single graph with this property, namely the Radograph. Thus any countably infinite random graph is almost surely the Radograph, which for this...
hereditarily finite sets, and defining the adjacency relation of the Radograph. In computer science, it is used for efficient representations of set...
proofs of the Erdős–Ko–Rado theorem and its analogue for intersecting families of subspaces over finite fields. For general graphs which are not necessarily...
such a sequence uniquely defines the Radograph.) He then defines Gi to be the induced subgraph of the Radograph formed by removing the final vertex (in...
countable infinite random graphs in the Erdős–Rényi model are, with probability 1, isomorphic to the highly symmetric Radograph. The smallest asymmetric...
complement graphs, and the Radograph. If a graph is 5-ultrahomogeneous, then it is ultrahomogeneous for every k. There are only two connected graphs that are...
there is a specific infinite graph, the Radograph R {\displaystyle R} , such that the sentences modeled by the Radograph are exactly the ones for which...
Many other symmetric graphs can be classified as circulant graphs (but not all). The Radograph forms an example of a symmetric graph with infinitely many...
\lambda _{0}} has multiplicity 1. The Erdős–Ko–Rado theorem states that the independence number of the Kneser graph K(n, k) for n ≥ 2 k {\displaystyle n\geq...
structure. In graph theory, the graph whose vertices correspond to hereditarily finite sets and edges correspond to set membership is the Radograph or random...
combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory...
Erdős–Ko–Rado theorem (which can be formulated in terms of independent sets in Kneser graphs), these are the unique maximum independent sets in this graph. It...
hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two...
logic of graphs. Moreover, the limiting probability is one if and only if the infinite Radograph has the property. For instance, a random graph in this...
particular any unclassifiable or deep theory, such as the theory of the Radograph. ℶ d + 1 ( | α + ω | ) {\displaystyle \beth _{d+1}(|\alpha +\omega |)}...
Johnson graphs are a special class of undirected graphs defined from systems of sets. The vertices of the Johnson graph J ( n , k ) {\displaystyle J(n...
Global Information