At this paper, on Section 2, it is constructed a set of $2^{n-2}$ points in the plane such that they do not contain a convex $n$-gon.
I am not able to grasp it. I get lost when it says:
Suppose now that $P_i, (i = 1, \dots,r)$ is a non-empty subset of $S_{k_i}, 1\leq k_1<\dots<k_r<n-1$ and such that $P=\cup_{i=1}^{r} P_i$ forms a convex polygon. Since the slope of lines within each $S_k$ is positive, $P_i$ for $1 < i < r$ consists of a single point and $P_1$ must form a concave sequence, $P_r$ a convex sequence. But then the total number of points in $P$ is at most $$k_1 + (k_r - k_1 - 1) + (n-k_r) = n-1$$
Seems to be really simple, but (i) where are the $2^{n-2}$ points derived from? (ii) how is derived the conclusion from the argument? Any clarification to help me understanding the proof would be really welcomed.
Thanks!
Edit
It has been detected by @aschepler that the paper linked is incomplete. A free link to the complete paper, or some site where the proof is showed, or an answer with the full proof would be welcomed.
