I am reading [a paper by Szekeres and Peters][1] on computing the 17-point case of [the Erdős–Szekeres conjecture][2]. The conjecture states that the minimum number of points in the plane (in general position, no three points collinear) such that any arrangement will contain a convex subset of $n$ points is
$$
2^{n-2}+1
$$

The paper describes an algorithm which tests if subset of points is convex by checking the orientation of each ordered triple of points. The signature function $\sigma$ maps an ordered triple of points $(a,b,c)$ to $\{+,-\}$, if the three points are oriented clockwise or counterclockwise respectively.

On page 6, the paper gives a convexity condition from 4 relations on the signatures of each ordered triple of points,

> Let abcde be any (ordered) set of five points of $S_9$. It forms a convex 5-subset if and
only if its Q = 10 triples satisfy one of the four relations (termed convex relations):
$$
R_1: \sigma(abc)=\sigma(bcd)=\sigma(cde)\\
\quad R_2: \sigma(abc) = \sigma(bce) = -\sigma(ade)\\
\quad R_3: \sigma(abd) = \sigma(bde) = -\sigma(ace)\\
\quad R_4: \sigma(acd) = \sigma(cde) = -\sigma(abe)
$$

And on page 9, there is a similar set of 8 relations for a 6-subset,

>Let abcdef be any (ordered) set of six points of $S_{17}$. It forms a convex 6-subset if and
only if its ${6 \choose 3} = 20$ triples satisfy one of the eight convex relations:
$$
R_1: \sigma(abc)=\sigma(bcd)=\sigma(cde)=\sigma(def)\\
\quad R_2: \sigma(abc) = \sigma(bcd) = \sigma(cdf) = -\sigma(aef) \\
\quad R_3: \sigma(abc) = \sigma(bce) = \sigma(cef) = -\sigma(adf) \\
\quad R_4: \sigma(abd) = \sigma(bce) = \sigma(def) = -\sigma(acf) \\
\quad R_5: \sigma(acd) = \sigma(cde) = \sigma(def) = -\sigma(abf) \\
\quad\quad R_6: \sigma(abc) = \sigma(bcf) = -\sigma(ade) = -\sigma(def) \\
\quad\quad R_7: \sigma(abd) = \sigma(bdf) = -\sigma(ace) = -\sigma(cef) \\
\quad\quad R_8: \sigma(acd) = \sigma(cdf) = -\sigma(abe) = -\sigma(bef) \\
$$

From these two quotes, it seems that there are a set of $2^{n-3}$ relations for $n$ point convexity. How are these convex relations derived? And how can the set of convex relations for an arbitrary number of points be found?

  [1]: https://www.cambridge.org/core/journals/anziam-journal/article/computer-solution-to-the-17point-erdosszekeres-problem/0EC7876789232266D60439A4C00D86D9
  [2]: https://en.wikipedia.org/wiki/Happy_ending_problem