# Geometric random graphs and Rado sets of continuous functions

@article{Bonato2018GeometricRG, title={Geometric random graphs and Rado sets of continuous functions}, author={Anthony Bonato and Jeannette C. M. Janssen and Anthony Quas}, journal={arXiv: Combinatorics}, year={2018} }

We prove the existence of Rado sets in the Banach space of continuous functions on [0,1]. A countable dense set S is Rado if with probability 1, the infinite geometric random graph on S, formed by probabilistically making adjacent elements of S that are within unit distance of each other, is unique up to isomorphism. We show that for a suitable measure which we construct, almost all countable dense sets in the subspaces of piecewise linear functions and of polynomials are Rado. Moreover, all…

## One Citation

Geometric Random Graphs on Circles

- MathematicsAdvances in Probability and Mathematical Statistics
- 2021

Given a dense countable set in a metric space, the infinite random geometric graph is the random graph with the given vertex set and where any two points at distance less than 1 are connected,…

## References

SHOWING 1-10 OF 20 REFERENCES

Geometric random graphs and Rado sets in sequence spaces

- Mathematics, Computer ScienceEur. J. Comb.
- 2019

It is shown that in the sequence spaces c and c_0, for measures $\mu$ satisfying certain conditions, almost all countable sets are Rado, and that representatives of the isomorphism classes obtained from the random geometric model with a randomly chosen dense countable vertex set are isomorphic to each other.

Infinite Random Geometric Graphs

- Mathematics
- 2009

We introduce a new class of countably infinite random geometric graphs, whose vertices V are points in a metric space, and vertices are adjacent independently with probability $${p \in (0, 1)}$$ if…

On random points in the unit disk

- Computer ScienceRandom Struct. Algorithms
- 2006

This work proves that almost always X ∼ n1−c2 when 0 ≤ c < 1 and shows that when λ = c √ lnn/n and c > 2, the diameter of G(λ, n) is bounded by (4 + o(1)/λ; and modify this construction to yield a function c(δ) > 0 such that the diameter is at most 2.

Highly connected random geometric graphs

- Mathematics, Computer ScienceDiscret. Appl. Math.
- 2009

It is shown that for s=o(logn), the threshold for s-connectivity is asymptotically the same as that for connectivity, so that, as the authors increase k, G"n","k becomes s-connected very shortly after it becomes connected.

Random geometric graphs.

- Medicine, MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2002

An analytical expression for the cluster coefficient is derived, which shows that the graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality are distinctly different from standard random graphs, even for infinite dimensionality.

The Random Graph Revisited

- Computer Science
- 2001

The random graph, or Rado's graph (the unique countable homogeneous graph) has made an appearance in many parts of mathematics since its first occurrences in the early 1960s and some old and new results on this remarkable structure are discussed.

A course on the Web graph

- Computer Science
- 2008

A Course on the Web Graph is the first mathematically rigorous textbook discussing both models of the web graph and algorithms for searching the web, and is based on a graduate course taught at the AARMS 2006 Summer School at Dalhousie University.

Approximate Isometries on Finite-Dimensional Normed Spaces

- Mathematics
- 1999

Every e-isometry between real normed spaces of the same finite dimension which maps the origin to the origin may be uniformly approximated to within 2e by a linear isometry. Under a smoothness…

The Random Graph

- Computer Science, MathematicsThe Mathematics of Paul Erdős II
- 2013

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

Random geometric graphs and isometries of normed spaces

- MathematicsTransactions of the American Mathematical Society
- 2018

<p>Given a countable dense subset <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S">
<mml:semantics>
<mml:mi>S</mml:mi>
…