Given $S\subseteq \mathbb{N}$, we associate to $S$ the *sum-graph* $G\_S = (\mathbb{N}, E)$ where $$E = \big\{\{m,n\}: m,n \in \mathbb{N} \text{ and } m+n\in S\big\}.$$


If $S$ is infinite, is $G\_S$ always connected?



Sum-graph over an infinite set
No, let $S=2\mathbb N$, the set of even numbers.


Then $2\mathbb N$ and $2\mathbb N+1$ (the set of odd nunbers) are two distinct connected components of $G\_S$. (Also, they are both complete subgraphs of $G\_S$.)


