This section of the article by E. Szemerédi discusses the problem of sets of integers that contain no arithmetic progressions of a given length \( k \). It begins with a review of van der Waerden's theorem, which states that any partition of the integers into two classes will contain an arithmetic progression of a certain length. The focus then shifts to the finite version of this theorem, where the goal is to find the smallest integer \( f(n) \) such that any partition of the integers from 1 to \( f(n) \) into two classes will contain an arithmetic progression of \( n \) terms. The best known upper bound for \( f(n) \) is extremely poor, while the best lower bound, due to Berlekamp, is \( f(n) < n^{2n} \) for prime \( n \). The section also covers the work of Erdős and Turán, who introduced the quantity \( r_k(n) \), defined as the largest integer \( l \) for which there exists a sequence of integers up to \( n \) that does not contain an arithmetic progression of \( k \) terms. They conjectured that \( r_k(n) < n^{1 - c_k} \) for some constant \( c_k \), and later Behrend proved that either \( c_k = 0 \) for all \( k \) or \( \lim_{k \to \infty} c_k = 1 \). Szemerédi's proof of the general conjecture \( c_k = 0 \) for all \( k \) is outlined, emphasizing the use of elementary combinatorial arguments and the concept of \( m \)-configurations, which are generalizations of arithmetic progressions. The proof involves showing that any set of integers with positive upper density must contain arbitrarily long arithmetic progressions.This section of the article by E. Szemerédi discusses the problem of sets of integers that contain no arithmetic progressions of a given length \( k \). It begins with a review of van der Waerden's theorem, which states that any partition of the integers into two classes will contain an arithmetic progression of a certain length. The focus then shifts to the finite version of this theorem, where the goal is to find the smallest integer \( f(n) \) such that any partition of the integers from 1 to \( f(n) \) into two classes will contain an arithmetic progression of \( n \) terms. The best known upper bound for \( f(n) \) is extremely poor, while the best lower bound, due to Berlekamp, is \( f(n) < n^{2n} \) for prime \( n \). The section also covers the work of Erdős and Turán, who introduced the quantity \( r_k(n) \), defined as the largest integer \( l \) for which there exists a sequence of integers up to \( n \) that does not contain an arithmetic progression of \( k \) terms. They conjectured that \( r_k(n) < n^{1 - c_k} \) for some constant \( c_k \), and later Behrend proved that either \( c_k = 0 \) for all \( k \) or \( \lim_{k \to \infty} c_k = 1 \). Szemerédi's proof of the general conjecture \( c_k = 0 \) for all \( k \) is outlined, emphasizing the use of elementary combinatorial arguments and the concept of \( m \)-configurations, which are generalizations of arithmetic progressions. The proof involves showing that any set of integers with positive upper density must contain arbitrarily long arithmetic progressions.