What set of points on the $256 \times 256$ integer grid maximizes the number of vertices in its convex hull?
For the full context, I was given an assignment to test algorithms that find the vertices of the convex hull of a set of points in $2D$. One natural thing to test is when we set the input set such that all the vertices (size $n$) are convex hull vertices (size $h$). In other words, the case when $n=h$. One major restriction was that the synthetic vertices I generate need to be integer pairs in $[0, 32767]^2$. To further clarify, co-linear points are not considered part of the convex hull (hence I want maximum number vertices). I quickly ran into issues because the trivial way of putting the vertices around the circumference of a circle no longer worked as I end up with something like here (range smaller here to highlight issue). Indeed, the maximum $h$ I could produce was very small (around $2000$).
I already talked with my professor and they told me it's possible to make $h \gg 2000$ but I don't personally believe it, so before I go to them again I'd like to know if it's possible to prove the fundamental limit to how large $h$ can get (an expression for a low upper bound).