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It is straightforward to check that the discrete cube $Q=\{0,1\}^n$ with $\ell^1$-metric is 3-point-homogeneous, but not 4-point-homogeneous (assuming $n$ is large). In other words, if $A\subset Q$ has at most 3 points, then any distance-preserving map $A\to Q$ can be extended to an isometry $Q\to Q$, but there is a 4-point set for which it does not hold.

I would like to see an example of $m$-point-homogeneous that is not $(m+1)$-point-homogeneous for $m\geqslant 4$.

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  • $\begingroup$ Do you mean finite metric spaces? $\endgroup$
    – Wlod AA
    Commented Sep 28, 2022 at 20:25
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    $\begingroup$ @WlodAA if you have an infinite one, then show it to me :) $\endgroup$
    – Anton Petrunin
    Commented Sep 28, 2022 at 20:53
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    $\begingroup$ (Everywhere I refer to a polytope in what follows I mean its vertices regarded as a metric space with the $\ell^2$ norm.) Some googling found a paper (arxiv.org/abs/2206.13096) showing that the $n$-orthoplex is $2n$-point homogeneous (Corollary 4). It also contains the example that both the regular $n$-gon and the $n$-simplex are $n$-point homogeneous but they are not "not $n+1$-point homogeneous" in your sense since they don't have $n+1$ points at all. I can't find any examples in the paper which are $2k-1$-point homogeneous but not $2k$-point homogeneous for $k \ge 3$. $\endgroup$
    – Qiaochu Yuan
    Commented Sep 29, 2022 at 3:38
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    $\begingroup$ @QiaochuYuan for a space with $<n$ points, the $n$-point homogeneity rather sounds like an empty condition, hence true (however to make things reasonable one should allow non-injective families, or equivalently require, in the definition of $n$-homogeneity, $n'$-homogeneity for all $n'<n$). So, a finite set with all points at distance $1$ should be $n$-homogeneous for all $n$. $\endgroup$
    – YCor
    Commented Sep 29, 2022 at 10:46
  • $\begingroup$ @QiaochuYuan right, for me all of them are all-set-homogeneous. $\endgroup$
    – Anton Petrunin
    Commented Sep 29, 2022 at 11:15

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The Schläfli graph, viewed as a metric space, is 4-homogeneous but not 5-homogeneous.

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