Asked by Erdős in 1987, according to Katznelson [Ka01]. In other words, does the Cayley graph defined on $\mathbb{Z}$ by a lacunary sequence have a finite chromatic number?

Katznelson observed that a positive solution to the problem follows from the answer to [464], which yields an irrational $\theta$ and $\delta>0$ such that $\inf_k \| \theta n_k\|>\delta$.

Indeed, given such a $\theta$ a colouring of $\mathbb{N}$ using $\ll \delta^{-1}$ colours lacking any solution to $a-b\in A$ can be produced by dividing $\mathbb{R}/\mathbb{Z}$ into disjoint intervals of length $\leq \delta$ and then colouring $n$ according to which interval $\| \theta n\|$ belongs to.

In particular, the solution to [464] implies the answer to this question is yes, with the best known quantitative bound, due to Peres and Schlag [PeSc10], being that there is a colouring with no solutions using at most \[\ll \epsilon^{-1}\log(1/\epsilon)\] colours.