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In the paper A Note on Triple Transitive Graphs, Cameron gives an almost-complete determination of triple transitive graphs with girth greater than $3$. In the remark at the end he says:

"If, as seems likely, the only graph occurring under (iv) is the Higman-Sims graph (with $\mu = 1$), then..."

Has this been resolved by now? Are there other triple transitive graphs of diameter $2$ that appear in (iv) of the classification?

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  • $\begingroup$ For what it's worth, Math Reviews gives no later reviews citing the Cameron paper. Also, searching MR for "triple transitive graphs" turned up nothing helpful. $\endgroup$
    – Gerry Myerson
    Apr 30, 2014 at 23:48
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    $\begingroup$ On the other hand, this paper predates the classification of finite simple groups and this seems like the kind of question where the classification could make a big difference (for example, the Higman-Sims graph involves a sporadic simple group, and, as noted in the sketch of the proof in the paper under question, a vertex-stabiliser acts 3-transitively on its neighbours, and 3-transitive groups were only classified post-classification, which might play a role.) Do you have access to reference [1] from the paper? That might be helpful, to see how the Higman-Sims graph comes up. $\endgroup$
    – verret
    May 1, 2014 at 3:00
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    $\begingroup$ More specifically, aren't graphs in (iv) rank 3 graphs, which have been classified? $\endgroup$
    – verret
    May 1, 2014 at 3:19
  • $\begingroup$ @verret, thanks. You are right, the graphs in (iv) are rank $3$, and indeed rank $3$ groups were classified post CFSG. Do you know of a reference for the classification of rank $3$ graphs? $\endgroup$
    – DKal
    May 1, 2014 at 6:50
  • $\begingroup$ I only knew that the groups of rank 3 were classified and assumed that the graphs of rank 3 also were. After failing to find a reference and asking around a bit, it seems that they havn't really been explicitly listed anywhere, although how to do it is clear, at least in principle. It may be that the extra information you have in your situation (for example the girth) would allow one to finish case iv) without findind all the rank 3 graphs. $\endgroup$
    – verret
    May 10, 2014 at 11:20

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