This is inspired by an older (as yet unanswered) question.

Let us call a set $S\subseteq\omega$ *thin in the 1st sense* if $$\lim\sup_{n\to\infty}\frac{|S\cap n|}{n+1}=0$$ where $\omega$ is the first infinite cardinal, and $n=\{0,\ldots,n-1\}$ for all $n\in\omega$ with $n>0$.

Moreover, to $S\subseteq \omega$ we associate a simple, undirected graph $G_S=(\omega, E_S)$ where $$E_S = \big\{\{a,b\}:a\neq b\in \omega \textrm{ and }a+b\in S\big\}.$$

Let us call $S\subseteq \omega$ *thin in the 2nd sense* if the chromatic number $\chi(G_S)$ is finite.

**Question.** Are there any implications between these two notions of thinness?