Structure $\left<X,B\right>$ is an $n$-dimensional Euclidean incidence space iff $B$ is ternary betweenness relation associated with $n$-dimensional Euclidean geometry. That is, Euclidean incidence space is Euclidean geometry minus the notions of congruences.

Let $n>2$. Do I conjecture correctly that $\left<X,B\right>$ is an $n$-dimensional Euclidean incidence space iff:

$(*)$ There is a matroid $\left<X,\mathcal S\right>$ of rank $n+1$. If $A \subseteq X$ is a flat of rank $k: 1 \leq k \leq n$ then $\left<A,B|A\right>$ is a $k-1$-dimensional Euclidean space.

I believe this to be the case looking at a way in which Hilbert axioms handle dimensions, with planes and lines axiomized extremely carefully, but axiomization of an entire $3$-dimensional space basically boiling down to stating something very similar to $(*)$.

If that was, indeed, a case, it would be a rather nice tool for checking whether or not something is a high-dimensional Euclidean incidence space.


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