Is it possible to construct an infinite vector of 0 and 1, i.e. an element of $x\in\{0,1\}^{\mathbb Z}$, with the following properties:
-there is a $K>0$ such that for every $\ell\in\mathbb Z$, $(x_i)_{i=\ell}^{\ell+K}$ contains both a 0 and a 1;
-$x$ does not contain any arithmetic progression of 0 or 1: i.e. for every $p,q\in\mathbb N,$ $(x_{p+nq})_{n\in\mathbb Z}$ contains both a 0 and a 1?