Given a finite indexing-set $I$ and a collection $P = \{P_i: \ i \in I\}$ of points in the plane no three of which are collinear, let $I_{(3)}$ denote the set of ordered triples of distinct elements of $I$, and let $f_P$ be the function from $I_{(3)}$ to $\{1,-1\}$ such that $f_P(i,j,k)$ is 1 (resp. $-1$) if the points $p_i,p_j,p_k$ lie in counterclockwise (resp. clockwise) order on the circle going through the three points. Call an $f$ that is of the form $f_P$ for some $P$ “achievable”. Is achievability a local condition, in the sense that there exists a fixed $k$ with the property that a function $f: I_{(3)} \rightarrow \{1,-1\}$ is achievable iff its restriction to $I’_{(3)}$ is achievable for all $k$-element subsets $I’ \subseteq I$?

The smallest unachievable $f$, with $|I|=4$, has $f(1,2,3)=f(1,4,2)=f(2,4,3)=f(3,4,1)$ (associated with the faces of a tetrahedron). To see why it can’t be achieved, note that the three lines through $P_1$, $P_2$, and $P_3$ divide the plane into seven regions; the specified $f$ would correspond to points in the eighth, nonexistent region.

This question is a sharpened version of my earlier question
https://mathoverflow.net/questions/160872/axiomatizing-orientation-in-the-complex-plane
somewhat in the spirit of the question https://mathoverflow.net/questions/50705/arrangements-of-points-in-the-plane .