A notion of thinness for subsets of $\omega$, using chromatic number

Question

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?

Answer 1

The two notions are incomparable.

To see that the first notion does not imply the second, let's construct a set $S$ with asymptotic density $0$, but with infinite chromatic number. We place infinitely many increasingly large intervals into $S$, but spaced very far apart, so that the density is zero. If $S$ has an interval centered at $n$ of size $k$, then all numbers within $k/2$ of $n/2$ will be connected to the others in your graph. This will cause a complete subgraph in $G_S$ of size $k$. Since $k$ becomes as large as desired, the chromatic number of $G_S$ will be infinite.

Conversely, to see that the second notion does not imply the first, consider the set $S$ of odd numbers, which has density $1/2$. If $a+b$ is odd, then the parities of $a$ and $b$ must differ. So every edge in $G_S$ connects an odd number with an even number, and never two even numbers or two odd numbers. So the graph is $2$-colorable.