Question involving an incidence geometry theorem from Larry Guth's book Polynomial Methods in Combinatorics [2016]

Question

At the very beginning of Chapter 11 of Larry Guth's book, we are given the following theorem which is supposed to be proved within the chapter:

Theorem 11.1. There is a constant K so that the following holds. If ℒ is a set of L lines in R^3 with |P_3(ℒ)| >= KL^(3/2), then there is a plane that contains at least 10L^(1/2) lines of ℒ.

Note that P_3(ℒ) is the set of 3-rich points of incidences in ℒ.

However we don't ever actually see any explicit proof of this result in the chapter and I was wondering if anybody here knows how to prove it. They end the chapter with a proof of this result:

Theorem 11.7. (Planar clustering theorem) There is a constant K so that the following holds. Let ℒ be a set of L lines in R^3 so that each line contains >= A = KL^(1/2) points of P_3(ℒ). Then ℒ lies in <= KL/A planes.

Can anyone use this Theorem 11.7 to prove Theorem 11.1? Doesn't look like Theorem 11.1 is even used anywhere else in the book but I'd still like to see a proof.

Answer 1

Yes, it can be proven using Theorem 11.7 and in fact it is almost done in the book.

In the Corollary 11.8 it is proven that

There exists constant $C$ s.t. the following holds. Suppose that $\mathfrak{L}$ is a set of $L$ lines in $\mathbb{R}^3$ that contains at most $B$ lines in any plane. If $B\ge L^{1/2}$, then $$|P_3(\mathfrak{L})|\le CBL.$$

From this the required theorem is obtained with $K>10C$, the proof is immediate by contradiction:

Suppose there is no such plane. Then Corollary 11.8 applies with $B=10L^{1/2}\ge L^{1/2}$. Then $|P_3(\mathfrak{L})|\le CBL<KL^{3/2}.$ which is contrary to the hypothesis.

Note that there is a typo in the statement of the Theorem 11.7 in the book, as seen in the proof of Corollary and later in the proof of the Theorem itself: it should be $ A \ge KL^{1/2} $ instead of equality.