Combinatorics of $k$-Farey graphs

With an eye towards studying curve systems on low-complexity surfaces, we introduce and analyze the $k$-Farey graphs $\mathcal{F}_k$ and $\mathcal{F}_{\leqslant k}$, two natural variants of the Farey graph in which we relax the edge condition to indicate intersection number $=k$ or $\le k$, respectively. The former, $\mathcal{F}_k$, is disconnected when $k>1$. In fact, we find that the number of connected components is infinite if and only if $k$ is not a prime power. Moreover, we find that each component of $\mathcal{F}_k$ is an infinite-valence tree whenever $k$ is even, and $\mathrm{Aut}(\mathcal{F}_k)$ is uncountable for $k>1$. As for $\mathcal{F}_{\leqslant k}$, Agol obtained an upper bound of $1+\min\{p:p\text{ is a prime}>k\}$ for both chromatic and clique numbers, and observed that this is an equality when $k$ is either one or two less than a prime. We add to this list the values of $k$ that are three less than a prime equivalent to $11\ (\mathrm{mod}\ 12)$, and we show computer-assisted computations of many values of $k$ for which equality fails.