Probability of random (in)finite graphs being isomorphic I once skimmed a paper which examined the probability of two (infinite) graphs picked at random being isomorph. The surprising result was that for two random infinite graphs this probability is quite high as opposed to the finite case. I remember it being referenced in the context of why so many graph invariants  - often claimed to be complete - work correct on most inputs.

I think the paper was published before the nineties.
From the comment: in the mathematical field of graph theory, the Rado graph [1], also known as the random graph or the Erdős–Rényi graph, is the unique (up to isomorphism) countable graph $R$ such that for every finite graph $G$ and every vertex $v$ of $G$, every embedding of $G − v$ as an induced subgraph of $R$ can be extended to an embedding of $G$ into $R$. As a result, the Rado graph contains all finite and countably infinite graphs as induced subgraphs.

In this note you can find other details, in particular: ... if we produce two random infinite graphs, then, with probability 1, they both have the extension property. Hence they are isomorphic (and also isomorphic to the Rado graph); in other words, any two random countably infinite graphs are isomorphic ... 

[1] Rado, Richard (1964), "Universal graphs and universal functions", Acta Arith. 9: 331–340.