In this chapter we continue the study of Szemerédi's theorem (Theorem 10.1), but now focus on the case of longer arithmetic progressions k > 3. While we have seen that the k = 3 case of this theorem can be treated by Fourier-analytic methods, it turns out that the higher-k case cannot be dealt with by (linear) Fourier-analytic tools, even when k = 4; we will see some justifications for this fact later. Indeed, whereas Roth's treatment [287] of the k = 3 case appeared in 1953, it was only in 1969 that Szemerédi [343] established the k = 4 case of Theorem 10.1, by combining the density increment argument of Roth with some impressively complicated combinatorial arguments (based on those discussed in Section 10.7). Unfortunately, this argument did not yield any new bound on van der Waerden's theorem (Exercise 6.3.7), as this theorem was used in the proof; note that one of the original motivation of Erdős and Turán in introducing this problem in [99] was to obtain a more effective bound on van der Waerden's theorem than the Ackermann-type bounds obtained by the usual proof methods. In 1972 Roth [288] obtained an alternative proof of the k = 4 case by combining the Fourier method with Szemerédi's arguments, but again van der Waerden's theorem was involved.

In 1975, Szemerédi [345] finally established Theorem 10.1 for all k. The argument is purely combinatorial. It uses the density increment argument, van der Waerden's theorem, and an induction on k, although to execute this induction properly, a number of new combinatorial tools needed to be introduced, most notably the very useful and influential Szemerédi regularity lemma, which has already been discussed in Section 10.6.