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Jun 9, 2020 at 12:25 comment added Gabe Conant The upper bound is tight (as suggested in Ilya Bogdanov's answer and observed earlier in the comments by Wojowu). If $D$ is the set of natural numbers not divisible by $k$ then $D$ has density $(k-1)/k$, and $G_D$ is $k$-colorable via the partition into equivalence classes modulo $k$.
Jun 9, 2020 at 0:20 comment added Capublanca Nice answer, thank you!
Jun 9, 2020 at 0:19 vote accept Capublanca
Jun 8, 2020 at 19:54 history answered Gabe Conant CC BY-SA 4.0