Problem 5

Prove that for every natural number

m

, there exists a finite set

S

of points in a plane such that every point

A

S

has exactly

m

points in

S

at unit distance from

A

Step 1 of 6: Recast the problem as regular unit-distance graphs

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S\text{ is }m\text{-regular}\iff\#\{B\in S:|A-B|=1\}=m\text{ for every }A\in S

Detailed analysis

Join two points of a finite set by an edge exactly when their Euclidean distance is $1$ . The required set is precisely a finite unit-distance graph in which every vertex has degree $m$ .