#discrete-geometry

Color every point of the plane so that any two points at distance exactly 1 receive different colors. What is the smallest number of colors for which this is possible? It is known that the answer is 5, 6 or 7. Determine it.

Let $f(N)$ be the smallest number such that any $f(N)$ points in the plane, no three on a line, include $N$ points that are the vertices of a convex polygon. Prove or disprove that $f(N)=2^{N-2}+1$ for every $N\ge 3$.

Prove that no packing of equal balls in three-dimensional space has average density greater than $\pi /\sqrt{18}\approx 0.7405$, the density of the familiar cannonball stacking (face-centered cubic packing).

Keller conjectured in 1930 that in every tiling of ${\mathbb{R}}^n$ by unit cubes, some two cubes share an entire $(n-1)$-dimensional face. For which dimensions $n$ is this true?

A finite set of points in the plane does not lie on a single line. Prove that some line passes through exactly two of the points.