Say that two lattice points $a$ and $b$ of $\mathbb{Z}^2$
are *visible* to one another if the line segment $ab$
contains no other lattice points.
While exploring lattice polygons all of whose vertices
are visible to one another, I noticed that it seems impossible
to have more than $4$ mutually visible lattice points.

. Prove that there is no set of $\ge 5$ distinct points of $\mathbb{Z}^2$ that are mutually visible to one another, or construct examples.Q1

I suspect this is elementary, but I'm not seeing a proof or a refutation.

. What is the higher-dimensional analog? What is the largest number $f(d)$ of mutually visible points of $\mathbb{Z}^d$ ?Q2