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 5 of 6: Exclude every unwanted cross-neighbor

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|(a-c)+e^{i\theta}(b-d)|^2=1

Detailed analysis

Any additional unit-distance pair would satisfy $|(a-c)+e^{i\theta}(b-d)|^2=1$ for some finite choices of $a,c,b,d$ . After expanding, this is a nontrivial equation of degree at most $2$ in $e^{i\theta}$ , so it excludes only finitely many rotations. Choose $\theta$ avoiding all of them; then no additional neighbors occur.