# Orientations of triples of points in the plane

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 somewhat in the spirit of the question Arrangements of points in the plane .

• "no two of which are collinear": Perhaps you mean no three of which are collinear? Apr 21, 2020 at 23:28
• I have replaced (geometry) by other tags, this tag is deprecated on MO - see the tag-info. I will also point out that you have linked to the Bjørn Kjos-Hanssen's answer rather than to the question - I am not sure whether that was intentional or it happened by mistake. Apr 21, 2020 at 23:39
• @Joseph: I did indeed mean three, not two, and have fixed the text accordingly. Thanks for catching that! Apr 22, 2020 at 1:04
• @Martin: That was an error on my part; thanks for catching it. I’ve replaced the link by the one I intended. Apr 22, 2020 at 1:07