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 arithmetic progressions, in the sense that all sufficiently large subsets contain at least one such progression. We prove that there exists a positive constant c and a set A⊂픽qn which does not contain a four‐term arithmetic progression, with the property that for every subset A′⊂A with |A′|≥|A|1−c, A′ contains a nontrivial three term arithmetic progression. We derive this from a more general quantitative Roth‐type theorem in random subsets of 픽qn, which improves a result of Kohayakawa–Łuczak–Rödl/Tao–Vu. We also discuss a similar phenomenon over the integers. Finally, we include another application of our methods, showing that for sets in 픽qn or ℤ the property of "having nontrivial three‐term progressions in all large subsets" is almost entirely uncorrelated with the property of "having large additive energy." [ABSTRACT FROM AUTHOR]