TITLE: Polygonal paths and polygons with prescribed set of vertices
QUESTION [7 upvotes]: Let $A$ be a finite set of points in the plane. How can we determine if there is a simple open polygonal path (i.e. without intersections),  whose vertices are exactly $A$, with no straight angles between adjacent sides?   Since in  mathoverflow.net/q/226469/4312 question is  only about cycles,  the stronger question also remains. Namely - how  to determine if there exists a polygon with non-intersecting sides and  without straight angles, whose vertices are exacly $A$ ?
Particular interesting case is when $A$ is a set of points $(x;y)$ with $0\leq x+y \leq 2n$ , where $x,y$ are nonnegative integers. Hypothesis : no for path for $n=1$ and $n=2$.

REPLY [5 votes]: The following polygon is a counterexample to the hypothesis for $n=4$. Namely we consider the set of integer  points $(x,y)|0\le x+y\le 8$. The picture is on a square-lined paper, where the size of one square is $\frac{1}{2}\times \frac{1}{2}$
The picture was constructed in collaboration with Svetlana Ermakova.
So my guess that it will be possible to do the same thing for all $n\ge 4$