Given a finite indexing-set $I$ and a collection $P = \{P_i: \ i \in I\}$ of points in the plane no two 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 $I’ \subseteq I$?

This question is a sharpened version of my earlier question Axiomatizing orientation in the complex plane somewhat in the spirit of the question Arrangements of points in the plane .