The Rado graph and the Urysohn space

Measure and category There are two natural ways of saying that a set of countable graphs is “large”. Choose a fixed countable vertex set, and enumerate the pairs of vertices: {x0, y0}, {x1, y1}, . . . There is a probability measure on the set of graphs, obtained by choosing independently with probability 1/2 whether xi and yi are joined, for all i. Now a set of graphs is “large” if it has probability 1. There is a complete metric on the set of graphs: the distance between two graphs is 1/2n if n is minimal such that xn and yn are joined in one graph but not the other. Now a set of graphs is “large” if it is residual in the sense of Baire category, that is, contains a countable intersection of open dense sets.