On a Ramsey–Turán variant of Roth's theorem

A classical theorem of Roth states that the maximum size of a solution‐free set of a homogeneous linear equation in is if and only if the sum of the coefficients of is 0. In this paper, we prove a Ramsey–Turán variant of Roth's theorem, with respect to a natural notion of “structured” sets introduced by Erdős and Sárközy in the 1970s. Namely, we show that the following statements are equivalent: Every solution‐free set of in with has size . There exists a non‐empty subset of coefficients of with zero sum.