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Statement

Item (c) (p. 277, quoted), the third of the aspects the paper lists as not examined. "Similar questions can be raised in other number theoretic contexts [Vardi $\ge$ 1998]. For example, consider the set of pairs of relatively prime integers in the plane connected if they are distance one apart: Does this set have a limiting density and if so, is it nonzero?"

The reference is I. Vardi, "Number theoretic percolation", listed as in preparation (p. 289). The print does not say which set's density is meant. The coprime pairs themselves have density $6/\pi^2$, so in the context of the paper's percolation questions the question presumably concerns the connected structure, such as an infinite component (a reading of this page).

Source. Ilan Vardi, Prime percolation, Experimental Mathematics 7 (1998), no. 3, 275--289, doi:10.1080/10586458.1998.10504373: p. 277, with the reference on p. 289. The edition read is identified on the source card.

Read depth. Claims checked: the passage was read on the printed page.

Bears on

• #1212: the question's graph has the problem's adjacency (two integer points at distance one differ by $\pm1$ in one coordinate) on the coprime pairs of the whole plane, not only of $\mathbb N^2$, and without the problem's conditions $\min(x,y)>1$ and a composite coordinate; it asks about density, not about a path to infinity. The paper proves nothing about it.