《图论与加性组合》第八章习题

Chapter 8 Sum-Product Problem

8.2 Crossing Number Inequality and Point-Line Incidences

Exercise 8.2.8 (Unit distance bound).
Using the crossing number inequality, prove given \(n\) points in the plane, at most \(O(n^{4/3})\) pairs of points are separated by exactly unit distance.

Proof. To prove such upper bound, we can first repeatedly delete points which are only incident to at most \(2\) points with unit distance. Therefore, we can assume in the point set \(\mathcal P\), every point has unit distance to at least \(3\) other points. Let \(V = \mathcal P\), and for each \(p\in \mathcal P\), consider the unit circle centered at \(p\), it pass through at least \(3\) points, we treat the arcs seperated by these points as edges in to be in \(E\). There must have multiple edges between two point, but the multiplicity is at most \(2\), thus \(2|E| \geq I\), where \(I\) is the pairs of points separated by unit distance.

Note that for each two unit circle, they intersects at most \(2\) times, we have \(\operatorname{cr}(G)\leq n^2\), by crossing number inequality, we have

\[n^2 \geq \operatorname{cr}(G) \gtrsim \frac{|E|^3}{n^2} \gtrsim \frac{I^3}{n^2}, \]

we can conclude that \(I \lesssim n^{4/3}\). \(\square\)