Four‐term progression free sets with three‐term progressions in all large subsets

This paper is mainly concerned with sets which do not contain four-term arithmetic progressions, but are still very rich in three-term the sense that all sufficiently large subsets at least one such progression. We prove there exists a positive constant c and set does progression, property for every subset , contains nontrivial three term derive this from more general quantitative Roth-type theorem random of improves result Kohayakawa–Łuczak–Rödl/Tao–Vu. also discuss similar phenomenon over integers. Finally, we include another application our methods, showing or “having progressions subsets” almost entirely uncorrelated additive energy.”