Pick two Erdös-Renyi (n,1/2) graphs uniformly at random. What's the chance they are the same (isomorphic)? Small. How small? Well, at most n!/ 2^(n choose 2). When n= 100, that's less than 10^-1300. OK, now let n=infinity. The chance that the two graphs are isomorphic is one (!). This is THE random graph (the Rado graph R). I will review its many non-random models and many strange properties. — It is a natural limit of the set of all finite graphs (a first order property is true for almost all finite graphs if and only if it holds with probability one in R) and this discontinuity is surprising.

In joint work with with Sourav Chatterjee we tried to find finite manifestations: For finite n, the largest isomorphic induced subgraph of a pair has size 4log (n) -2loglog(n)-2log(4/e) +1 (within 1, all logs base 2 in probability when n is large). This matches data amazingly well (e.g. for n more than 30) and illuminates problems in constraint satisfaction.