Incidence geometry studies which points lie on which curves, lines or higher-dimensional objects and how many such incidences a finite configuration can have.
For lines and points in the real plane, the number of point-line incidences isConnecting consecutive incidence points on each line and applying the Crossing lemma proves the bound.
For distinct real univariate polynomials of degree at most and points with distinct first coordinates, the number of incidences between the points and the polynomial graphs isTwo polynomial graphs meet at most times, so the graph formed from consecutive incidences has crossings; the Crossing lemma supplies the lower bound.
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Incidence geometry is a branch of geometry that focuses on the relationships and properties involving points and lines (or more generally, sets of geometric objects) without necessarily defining distances, angles, or other constructs commonly used in Euclidean geometry. It primarily studies the rules dictating how points, lines, and other geometric entities interact in terms of incidence, which refers to the notion of whether certain points lie on certain lines or if certain lines intersect.