We begin our study of geometric incidences by surveying the field and deriving a few first bounds. In this chapter we only discuss classical discrete geometry, from before the discovery of the new polynomial methods. This makes the current chapter rather different than the rest of the book (outrageously, it even includes some graph theory). We also learn basic tricks that are used throughout the book, such as double counting, applying the Cauchy–Schwarz inequality, and dyadic decomposition.

Topics that are discussed in this chapter: the Szemerédi–Trotter theorem, a proof of this theorem that relies on the crossing lemma, the unit distances problem, the distinct distances problem, a problem about unit area triangles, the sum-product problem, rich point, point-line duality.

In Chapter 7 we studied the ESGK framework. This was a reduction from the distinct distances problem to a problem about pairs of intersecting lines in R^3. In the current chapter we further reduce the problem to bounding the number of rich points of lines in R^3. We solve this incidence problem with a more involved variant of the constant-degree polynomial partitioning technique. This completes the proof of the Guth–Katz distinct distances theorem.

The original proof of Guth and Katz is quite involved. We study a simpler proof for a slightly weaker variant of the distinct distances theorem. This simpler proof was introduced by Guth and avoids the use of tools such as flat points and properties of ruled surfaces.