Abstract
We construct uncountable graphs in which any two isomorphic subgraphs of size at most 3 can be carried one to the other by an automorphism of the graph, but in which some isomorphism between 2-element subsets does not extend to an automorphism. The corresponding phenomenon does not occur in the countable case. The construction uses a suitable construction of infinite homogeneous coloured chains.
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Droste, M., Giraudet, M. & Macpherson, D. Set-Homogeneous Graphs and Embeddings of Total Orders. Order 14, 9–20 (1997). https://doi.org/10.1023/A:1005880810385
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DOI: https://doi.org/10.1023/A:1005880810385