Abstract We discuss the papers P. Erdős and P. Turán, On some sequences of integers , J. London Math. Soc. (1) 11 (1936), 261–264 and K. F. Roth, On certain sets of integers , J. London Math. Soc. (1) 28 (1953), 104–109, both foundational papers in the study of arithmetic progressions in sets of integers, and their subsequent influence.
Ben Green (Sun,) studied this question.