- The unit distance problem in the plane asks for the maximum number $U(n)$ of unit distances which can be obtained by $n$ points.
- For $k$ unit circles and $m$ points in the plane, $I(k,m)$ counts the number of point-circle incidences.
Now, in Matousek's Discrete Geometry book, he claims that $U(n) \leq \frac{1}{2} I(n,n)$, but my questions is shouldn't it be $U(n) \leq \frac{1}{2} I(n,2n)$, since any two circles possibly intersect in two points?
$I(n,n)$ is correct: Take $n$ points realizing $U(n)$ unit distances and draw a unit circle around each. Each pair of points corresponds to two unit circles, which gives you two point-circle incidences. In other words, it takes two points in the unit-distance problem to produce two incidences in the point-circle problem, so the number of points in the $U$ function and the $I$ function should be the same.