Problem #97: Does every convex polygon have a vertex with no other $4$...

---

Progress

Erdős originally conjectured this (in [Er46b]) with no $3$ vertices equidistant, but Danzer found a convex polygon on 9 points such that every vertex has three vertices equidistant from it (but this distance depends on the vertex). Danzer's construction is explained in [Er87b]. Fishburn and Reeds [FiRe92] have found a convex polygon on 20 points such that every vertex has three vertices equidistant from it (and this distance is the same for all vertices).

If this fails for $4$, perhaps there is some constant for which it holds? In [Er75f] Erdős claimed that Danzer proved that this false for every constant - in fact, for any $k$ there is a convex polygon such that every vertex has $k$ vertices equidistant from it. Since this claim was not repeated in later papers, presumably Erdős was mistaken here.

Erdős suggested this as an approach to solve [96]. Indeed, if this problem holds for $k+1$ vertices then, by induction, this implies an upper bound of $kn$ for [96].

The answer is no if we omit the requirement that the polygon is convex (I thank Boris Alexeev and Dustin Mixon for pointing this out), since for any $d$ there are graphs with minimum degree $d$ which can be embedded in the plane such that each edge has length one (for example one can take the $d$-dimensional hypercube graph on $2^d$ vertices). One can then connect the vertices in a cyclic order so that there are no self-intersections and no three consecutive vertices on a line, thus forming a (non-convex) polygon.