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?
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asked Apr 8, 2015 at 8:39
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1 Answer
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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 $G_S$.)
answered Apr 8, 2015 at 8:53