Arithmetic progressions of length three in subsets of a random set

Abstract

0. Introduction. In 1936 Erdős and Turan [ET 36] asked whether for every natural number k and every positive constant α, every subset A of [n] = {0, 1, . . . , n − 1} with at least αn elements contains a k-term arithmetic progression provided n is sufficiently large with respect to α and k. This conjecture was resolved by Roth [Ro 53] for k = 3, whereas for general k it was settled in the affirmative by the outstanding theorem of Szemeredi [Sz 75]. A few years after Szemeredi’s paper was published, an entirely different proof of this result, based on ergodic theory, was given by Furstenberg [Fu 77]. Since then, the main open problem concerning the original question of Erdős and Turan has been to find better lower bounds for the size of A that guarantee the existence of arithmetic progressions of length k in A. Unfortunately, not much has been accomplished for k ≥ 4. The explicit estimates that follow from Szemeredi’s original proof are very poor and Furstenberg’s approach does not provide such bounds at all. The case k = 3 is much better understood. Roth’s original argument implies that it is enough to assume that |A| ≥ n/ log log n and the best lower bound to date has been given independently by HeathBrown and Szemeredi (see [H-B 87]), who showed that for some absolute constant c > 0 every subset of [n] with at least n/(log n) elements contains an arithmetic progression of length three, provided n is sufficiently large.