Given $S\subseteq \mathbb{N}$, we associate to $S$ the *sum-graph* $G\_S = (\mathbb{N}, E)$ where $$E = \big\{\{m,n\}: m,n \in \mathbb{N} \text{ and } m+n\in S\big\}.$$
Is there an infinite subset $S\subseteq\mathbb{N}$ such that $G\_S$ has infinitely many connected components?



Connected components of a sum-graph over an infinite set
Write $S=\{s\_1, s\_2, \ldots\}$ with $s\_i<s\_{i+1}$ for all $i$. Then $G\_S$ has infinitely many connected components as soon as $s\_{i+1}>2s\_i$ for all $i$. (In particular, $S=\{3^k: k\in \mathbf{N}\}$ will do.)


To see this, consider the induced subgraph $\Gamma\_{n+1}$ on the vertices $\{1,\ldots,s\_{n+1}\}$. It is obtained from the induced subgraph $\Gamma\_n$ on the vertices $\{1,\ldots,s\_n\}$ by "mirroring" at $s\_{n+1}/2$. The connected components of $\Gamma\_n$ now get twice as big and stay disjoint, while there is at least one new connected component due to the condition $s\_{i+1}>2s\_i+1$. 
The claim now follows by induction.


