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Mar 27, 2018 at 13:56 history edited Peter Heinig CC BY-SA 3.0
Corrected a technical term.
Mar 27, 2018 at 13:55 comment added Peter Heinig It's worth pointing out that (0) the first published proof of this [C. Ward Henson, A family of countable homogeneous graphs, Pacific Journal of Mathematics, Vol. 38, No. 1, 1971], and (1) the technical term for the extension property you mentioned (which evidently is a strengthening of vertex-transitivity, which after all can be seen as extenting a partial isomorphism between 1-vertex-subgraphs) is ultrahomogeneous (Henson called it 'homogeneous', but that usually means something weaker: that there is some automorphism mapping one of the isomorphic subgraphs to the other).
Mar 27, 2018 at 13:15 vote accept Pablo
Mar 27, 2018 at 9:34 history edited Fedor Petrov CC BY-SA 3.0
added 9 characters in body
Mar 27, 2018 at 9:25 history answered Fedor Petrov CC BY-SA 3.0