Danzer has found a convex polygon on 9 points such that every vertex has three vertices equidistant from it (but this distance depends on the vertex), and Fishburn and Reeds 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?
This is false; Kovač [Ko23] provides an explicit (and elegantly simple) colouring using 25 colours such that no colour class contains the vertices of a rectangle of area $1$. The question for parallelograms remains open.
In fact, such a set does exist, as proved by Jackson and Mauldin [JaMa02]. Their construction depends on the axiom of choice.