Are the theories of betweenness the same in $\mathbb{R}^n$ for all $n\geq 2$?

Question

Consider $\mathbb{R}^n$, for some $n$ greater than or equal to $2$. We can form a structure by adjoining to it the ternary betweenness relation $B(x,y,z)$. Are all those structures elementarily equivalent? I am excluding $\mathbb{R}^1$, because in that structure, all points are collinear.

Answer 1

No. Using betweenness, you can express what it means for three points to be collinear (one is between the other two). Then, given a $n$-tuple of points, you can define the affine space they generate, namely all points you can generate by repeatedly taking collinear points up to $n-1$ times (for instance, for $n=3$, you first take a new point collinear with two of the points, and then take a point collinear with the new point and the third point). So you can detect the dimension as the least $n$ such that there is an $(n+1)$-tuple of points that affinely generates the whole space.

Answer 2

Here is a sentence separating $\mathbb{R}^2$ from $\mathbb{R}^n$ for $n>2$:

For any five points, no three of which are collinear, there are four points $A,B,C,D$ among those five such that the line segments $AB$ and $CD$ intersect.

I hope it is clear how to express this as a first-order sentence in the language of betweenness. If not, let me know which part you're having trouble with.

The point is that for any five points in the plane, no three of which are collinear, four of them form a convex quadrilateral, the diagonals of which intersect. In higher dimensions, it is easily arranged that no four points among the five are coplanar.