it is a classical theorem of roth that every dense subset of $\left\{1,\ldots,n\right\}$ contains a nontrivial three-term arithmetic progression. quantitatively, results of sanders, bloom, and bloom-sisask tell us that subsets of relative density at least $1/(\log n)^{1-\epsilon}$ already have this property. in this talk, we will discuss about some sets of $n$ integers which unlike $\left\{1,\ldots,n\right\}$ do not contain nontrivial four-term arithmetic progressions, but which still have the property that all of their subsets of density at least $1/(\log n)^{1-\epsilon}$ must contain a three-term arithmetic progression. perhaps a bit surprisingly, these sets turn out not to have as many three-term progressions as one might be inclined to guess, so we will also address the question of how many three-term progressions can a four-term progression free set may have. finally, we will also discuss about some related results over $\mathbb{f}_{q}^n$. based on joint works with jacob fox and oliver roche-newton.