Edge-disjoint double rays in infinite graphs: a Halin type result

We show that any graph that contains k edge-disjoint double rays for any k>0 contains also infinitely many edge-disjoint double rays. This was conjectured by Andreae in 1981.Comment: 15 pages, 2 figure

The random graph

Erd\H{o}s and R\'{e}nyi 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.Comment: Revised chapter for new edition of book "The Mathematics of Paul Erd\H{o}s

On Andreae's Ubiquity Conjecture

A graph $H$ is ubiquitous if for every graph $G$ that for every natural number $n$ contains $n$ vertex-disjoint $H$-minors contains infinitely many vertex-disjoint $H$-minors. Andreae conjectured that every locally finite graph is ubiquitous. We give a disconnected counterexample to this conjecture. It remains open whether every connected locally finite graph is ubiquitous