FI-MO 2009 Problem 4

We say that the set of step lengths $D\subset \mathbb{Z}_+=\{1,2,\ldots\}$ is excellent if it has the following property: If we split the set of integers into two subsets $A$ and $\mathbb{Z}\setminus{A}$ , at least other set contains element $a-d,a,a+d$ (i.e. $\{a-d,a,a+d\} \subset A$ or $\{a-d,a,a+d\}\in \mathbb{Z}\setminus A$ from some integer $a\in \mathbb{Z},d\in D$ .) For example the set of one element $\{1\}$ is not excellent as the set of integer can be split into even and odd numbers, and neither of these contains three consecutive integer. Show that the set $\{1,2,3,4\}$ is excellent but it has no proper subset which is excellent.