TITLE: Visualizing lower bound for the Happy Ending Problem (small cases).
QUESTION [5 upvotes]: 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.

Caption: Examples for $s=4$ (violet) with $4$ points for and $s=5$ (green) with $8$ points.

REPLY [2 votes]: 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:

   1      1
   2301   -459
   5305  -1210
   5306  -1208
   5307  -1201
   5308  -1170
   6592  -1639
   6593  -1630
   6594  -1628
   6595  -1567
   6596  -1565
   6597  -1558
   6938  -1812
   6939  -1764
   6940  -1755
   6941  -1753

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