I'm studying the Happy Ending Problem now, which states that
For any positive integer $s$, any sufficiently large finite set of points in the plane in general position has a subset of $s$ points that form the vertices of a convex polygon.
Let $g(s)$ denote the minimum number of points in general position whose set must contain a convex $s-$gon. It's known that $g(s) = 2^{s-2}+1$ for $s\le 6$, but the exact value for $s>6$ is unknown. However, it has been shown that $g(s) > 2^{s-2}$ for all $s$ through an example of a set of $2^{s-2}$ points without any convex $s-$gon. I've seen the construction of this example, but I have a hard time visualizing it because it involves organizing sets of points very distant from each other recursively.
I'm looking for a geometric visualization of sets like that for small cases of $s$. It is easy to find the setting for $s = 4$ and $s = 5$ (which are as follows) in the online articles on the subject, but I was unable to find any illustration for $s\ge 6$. An example for the case $s = 7$ would satisfy me a lot.
Above, examples for $s=4$ (violet) with $4$ points for and $s=5$ (green) with $8$ points.
Edit: Going with my intuition, I have found the following set of $16$ points which seems to not contain any convex hexagon. There are too many polygons to check, though, despite of all the symmetries, so I'm not sure about it.
In the figure I'm using a polar grid with an angle of $\dfrac{\pi}{40}$, in case someone wants to extract the exact coordinates.
I think those examples for $4\le s\le 6$ (if the last one is correct) consisting of concentric $s$-gons are pretty neat, but my hopes for this pattern to continue are not very high.
If you can track down a copy of the paper "On some extremum problems in elementary geometry" by Erdős and Szekeres (Ann. Univ. Sci. Budapest. Eotvos Sect. Math. 3-4 (1961) 53-63.), they describe such a construction of $2^{n-2}$ set that is void of convex $n$-gons. It is also reprinted in "Paul Erdős: the art of counting. Selected writings". Alas, there is no visualization there in the paper but an algorithm to construct such sets. I don't have copies of these on hand, but you may try to find them. Best of luck!
Edit.
If I didn't implement it incorrectly, we have for $n=6$ the following $16$ points in cartesian coordinates without a convex hexagon:
Which is kind of hard to see from its plot:
And for $n=7$ the following $32$ points:
1 1 16135 -2688 63595 -12180 63596 -12178 63597 -12171 63598 -12140 63599 -11975 96151 -20319 96152 -20310 96153 -20308 96154 -20247 96155 -20245 96156 -20238 96157 -19499 96158 -19497 96159 -19490 96160 -19459 109624 -24810 109625 -24762 109626 -24753 109627 -24751 109628 -23990 109629 -23981 109630 -23979 109631 -23918 109632 -23916 109633 -23909 111766 -25881 111767 -25581 111768 -25533 111769 -25524 111770 -25522