SEMINAR: Mathematics & Statistics Colloquium
Everyone is warmly invited to our Mathematics and Statistics Colloquium.
Talk title: The random graph and its friends
Talk abstract: There is a countably infinite graph R (first explicitly constructed by
Richard Rado) with the following remarkable property: If we choose a
countable random graph by selecting edges independently with probability
1/2, then with probability 1 it is isomorphic to R. (This fact was implicit in
a paper of Erdos and Renyi at about the same time as Rado's
construction.) The graph has many other surprising properties, and
occurs in a number of guises.
It turns out that the graph is produced by a construction by
Fraisse more than ten years earlier, which builds homogeneous
relational structures with prescribed finite
substructures, and shows its uniqueness. But even Fraisse had
been anticipated by Urysohn, who showed (in a posthumous paper a quarter
of a century earlier) that there is a unique homogeneous Polish space
containing all finite metric spaces.
It is natural to ask what happens if we dualise Fraisse's
construction by turning the arrows around. There is indeed a dual
construction; among other things it gives a new way to build a
remarkable topological space, the pseudo-arc.
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