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 which does not contain a four-term arithmetic progression, with the property that for every subset with , contains a nontrivial three term arithmetic progression. We derive this from a more general quantitative Roth-type theorem in random subsets of , 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 or the property of “having nontrivial three-term progressions in all large subsets” is almost entirely uncorrelated with the property of “having large additive energy.”

中文翻译：

---

在所有大型子集中具有三项级数的四项无级数集

本文主要关注不包含四项算术级数但仍然非常丰富的三项算术级数的集合，因为所有足够大的子集都包含至少一个这样的级数。我们证明存在一个正常数c和一个不包含四项算术级数的集合，其性质是对于每个具有 的子集，都包含一个非平凡的三项算术级数。我们从随机子集中的更一般的定量罗斯型定理推导出这一点，这改进了 Kohayakawa–Łuczak–Rödl/Tao–Vu 的结果。我们还讨论了整数上的类似现象。最后，我们包括了我们方法的另一个应用，表明对于集合或“在所有大子集中具有非平凡的三项级数”的特性几乎与“具有大的附加能量”的特性完全不相关。