This paper discusses the problem of finding sets of integers that do not contain arithmetic progressions of a given length. In 1926, van der Waerden proved that any partition of the integers into two classes must contain an arbitrarily long arithmetic progression in at least one class. However, the exact value of the function f(n), which gives the smallest number such that any partition of the integers from 1 to f(n) into two classes contains an arithmetic progression of length n, is still not well understood. Erdős and Turán studied the function r_k(n), which is the largest size of a subset of {1, ..., n} that does not contain an arithmetic progression of length k. They conjectured that r_k(n)/n approaches zero as n increases, which would imply that f(k) is small. Behrend showed that either c_k = 0 for all k or c_k approaches 1 as k increases. Later, Roth proved that r_3(n) is bounded by cn/log log n, and that r_4(n) is o(n). The author proved that r_k(n) = o(n) for all k, using van der Waerden's theorem. The paper outlines a proof of Erdős and Turán's conjecture that c_k = 0 for all k. The proof uses a key lemma that any finite graph can be partitioned into few nearly regular subgraphs. The main idea is that if a set of integers has positive upper density, then it must contain arbitrarily long arithmetic progressions. The proof involves generalizations of arithmetic progressions called m-configurations, and shows that a set with positive density must contain long arithmetic progressions. The paper concludes with a brief outline of the proof.This paper discusses the problem of finding sets of integers that do not contain arithmetic progressions of a given length. In 1926, van der Waerden proved that any partition of the integers into two classes must contain an arbitrarily long arithmetic progression in at least one class. However, the exact value of the function f(n), which gives the smallest number such that any partition of the integers from 1 to f(n) into two classes contains an arithmetic progression of length n, is still not well understood. Erdős and Turán studied the function r_k(n), which is the largest size of a subset of {1, ..., n} that does not contain an arithmetic progression of length k. They conjectured that r_k(n)/n approaches zero as n increases, which would imply that f(k) is small. Behrend showed that either c_k = 0 for all k or c_k approaches 1 as k increases. Later, Roth proved that r_3(n) is bounded by cn/log log n, and that r_4(n) is o(n). The author proved that r_k(n) = o(n) for all k, using van der Waerden's theorem. The paper outlines a proof of Erdős and Turán's conjecture that c_k = 0 for all k. The proof uses a key lemma that any finite graph can be partitioned into few nearly regular subgraphs. The main idea is that if a set of integers has positive upper density, then it must contain arbitrarily long arithmetic progressions. The proof involves generalizations of arithmetic progressions called m-configurations, and shows that a set with positive density must contain long arithmetic progressions. The paper concludes with a brief outline of the proof.