# The Random Graph

@inproceedings{Cameron2013TheRG, title={The Random Graph}, author={Peter J. Cameron}, booktitle={The Mathematics of Paul Erdős II}, year={2013} }

Erdős and Renyi showed the paradoxical result that there is a unique (and highly symmetric) countably infinite random graph. This graph, and its automorphism group, form the subject of the present survey.

## Topics from this paper

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## References

SHOWING 1-10 OF 85 REFERENCES

The random graph has the strong small index property

- Computer Science, MathematicsDiscret. Math.
- 2005

It is shown that the countable random graph has the small index property and a general theorem about permutation groups is deduced and the automorphism group of the random graph is not isomorphic to the Automorphism Group of any other countable homogeneous graph or digraph.

A Theorem on Reconstruction of Random Graphs

- Mathematics, Computer ScienceComb. Probab. Comput.
- 1993

In this paper we prove that given a finite collection of finite graphs, and the subsets of vertices of a random graph G that induce those graphs, it is almost always possible to uniquely reconstruct…

A Problem of Ulam on Planar Graphs

- Computer Science, MathematicsEur. J. Comb.
- 1981

The following problem was raised by S. M. Ulam: Does there exist a countable planar graph Go such that every countable planar graph is isomorphic to a subgraph of G 0 ? We answer this question in the…

Overgroups of the Automorphism Group of the Rado Graph

- Mathematics
- 2013

We are interested in overgroups of the automorphism group of the Rado graph. One class of such overgroups is completely understood; this is the class of reducts. In this article we tie recent work on…

A New Strongly Minimal Set

- Mathematics, Computer ScienceAnn. Pure Appl. Log.
- 1993

A new class of ℵ 1 categorical structures is constructed, disproving Zilber's conjecture, and some of their properties are studied.

Countable homogeneous partially ordered sets

- Mathematics
- 1979

It is shown that there are only countably many countable homogeneous partially ordered sets, thereby affirming a conjecture of Henson [2]. A classification of these partially ordered sets is given.

Reducts of Random Hypergraphs

- Mathematics, Computer ScienceAnn. Pure Appl. Log.
- 1996

It is shown that there exist only finitely many closed permutation groups G such that AUt(rk ) < G < &vn( rk) and each of the associated reducts of rk is homogeneous with respect to a finite relational language.

Random Structures and Zero-One Laws

- Mathematics
- 1993

Random structures (such as the classical random graphs of Erdős and Renyi) are playing an increasingly large role in the theory of computing, as well as in discrete mathematics. The surprising and…

The Small Index Property for ω‐Stable (ω‐Categorical Structures and for the Random Graph

- Mathematics
- 1993

We give a criterion involving existence of many generic sequences of automorphisms for a countable structure to have the small index property. We use it to show that (i) any a>-stable co-categorical…

Some isometry groups of the Urysohn space

- Mathematics, Computer ScienceAnn. Pure Appl. Log.
- 2006

Various isometry groups of the Urysohn space are constructed, including abelian groups which act transitively, and free groups which are dense in the full isometry group.