Given that simply stipulating positive upper density is not sufficient to guarantee that all but finitely many members are in an arithmetic progression of length 3, that there indeed exists sets of integers with positive density that contain infinitely many elements that are not contained in an arithmetic progression of length 3, are there counterexamples of (upper) density arbitrarily close to 1? In other words, let $0 < \epsilon < 1/2$ be given. Does there exist a set $A$ of positive integers such that the density of $A$ is larger than $1 - \epsilon$ and $A$ contains infinitely many elements that are not in an arithmetic progression of length 3?
Given that simply stipulating positive upper density is not sufficient to guarantee that all but finitely many members are in an arithmetic progression of length 3, that there indeed exists sets of integers with positive density that contain infinitely many elements that are not contained in an arithmetic progression of length 3, are there counterexamples of (upper) density arbitrarily close to 1? In other words, let $0 < \epsilon < 1/2$ be given. Does there exist a set $A$ of positive integers such that the density of $A$ is larger than $1 - \epsilon$ and $A$ contains infinitely many elements that are not in an arithmetic progression of length 3?