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
[Axiomatizing orientation in the complex plane](https://mathoverflow.net/questions/160872/axiomatizing-orientation-in-the-complex-plane)
somewhat in the spirit of the question [Arrangements of points in the plane](https://mathoverflow.net/questions/50705/arrangements-of-points-in-the-plane) .



Orientations of triples of points in the plane
If I understand your function correctly, this is the so called „order type“ of a point set, introduced by Goodman and Pollack, see e.g. [this survey](https://link.springer.com/chapter/10.1007/978-3-642-58043-7_6), which also contains the references to everything that I mention in the following. The question is now whether there is a number k s.t. if for order type of size n every partial order type of size k is realizable, then the whole order type is realizable.


My short answer: probably not


The slightly longer version:
I believe I have seen a construction of a non-realizable order type of size n where every partial order type of size n-1 is realizable. However, I dis not find this construction anymore, so I might be confusing it with a different setting. If I find it later, I will update my answer.


There are also other reasons for my „probably not“-answer. The first is that deciding whether an order type is realizable is NP-hard as shown by Shor (in fact, Mnëv has shown that it is ETR-hard, that is, the question whether a system of polynomial equations and inequalities has a solution in the reals is reducible in polynomial time to the question whether an irder type is realizable). If the above number k would exist, it would imply a polynomial time algorithm for order type realizability, proving P=NP=ETR.


There is also the related setting of allowable sequences, which is the setting in yor second related question [Arrangements of points in the plane](https://mathoverflow.net/questions/50705/arrangements-of-points-in-the-plane "Arrangements of points in the plane"). In this setting, there is an example of a configuration that is not realizable, but every subconfiguration is, see Theorem 2.1 and Figure 2.3 in the survey by Goodman and Pollack. If you allow for collinearities (taking the value 0 in your function), this construction can be adapted to order types by placing additional points at the intersections of the „diagonals“.


