A new paper on the topic has appeared, and while it does not answer your question I find the result interesting enough to be mentioned in a separate answer.

DeCorte and Golubev prove there that for $r$ large enough ($r\ge 12$ suffices), the distance-$r$ graph of the hyperbolic plane with curvature $\kappa=-1$ has measurable chromatic number at least $6$ (here "measurable" means that the maps sending a point to its color is Lebesgue Measurable). It is known for long that the measurable chromatic point of the plane is at least $5$ (Falconer 1981), so the interesting point here is that the lower bound gets better for large distances (i.e. for very negative curvature, if one prefers to fix $r=1$ and let $\kappa$ run over $(-\infty,0)$).

Note that under the Solovay axioms, the measurable hypothesis can be dropped since all sets are Lebesgue measurable (but then we don't have the full axiom of choice, and thus no DeBruin-Erdös theorem: we cannot deduce that under the Solovay axioms there is a finite unit-distance graph whose chromatic number is at least $6$).

A new paper on the topic has appeared, and while it does not answer your question I find the result interesting enough to be mentioned in a separate answer.

DeCorte and Golubev prove there that for $r$ large enough ($r\ge 12$ suffices), the distance-$r$ graph of the hyperbolic plane with curvature $\kappa=-1$ has measurable chromatic number at least $6$ (here "measurable" means that the maps sending a point to its color is Lebesgue Measurable). It is known for long that the measurable chromatic point of the plane is at least $5$ (Falconer 1981), so the interesting point here is that the lower bound gets better for large distances (i.e. for very negative curvature, if one prefers to fix $r=1$ and let $\kappa$ run over $(-\infty,0)$).

Note that under the Solovay axioms, the measurable hypothesis can be dropped since all sets are Lebesgue measurable (but then we don't have the full axiom of choice, and thus no DeBruin-Erdös theorem: we cannot deduce that under the Solovay axioms there is a finite unit-distance graph whose chromatic number is at least $6$).

A new paper on the topic has appeared, and while it does not answer your question I find the result interesting enough to be mentioned in a separate answer.

DeCorte and Golubev prove there that for $r$ large enough ($r\ge 12$ suffices), the distance-$r$ graph of the hyperbolic plane with curvature $\kappa=-1$ has measurable chromatic number at least $6$ (here "measurable" means that the maps sending a point to its color is Lebesgue Measurable). It is known for long that the measurable chromatic point of the plane is at least $5$ (Falconer 1981), so the interesting point here is that the lower bound gets better for large distances (i.e. for very negative curvature, if one prefers to fix $r=1$ and let $\kappa$ run over $(-\infty,0)$).

Note that under the Solovay axioms, the measurable hypothesis can be dropped since all sets are Lebesgue measurable (but then we don't have the full axiom of choice, and thus no DeBruin-Erdös theorem: we cannot deduce that under the Solovay axioms there is a finite unit-distance graph whose chromatic number is at least $6$).

A new paper on the topic has appeared, and while it does not answer your question I find the result interesting enough to be mentioned in a separate answer.

DeCorte and Golubev prove there that for $r$ large enough ($r\ge 12$ suffices), the distance-$r$ graph of the hyperbolic plane with curvature $\kappa=-1$ has measurable chromatic number at least $6$ (here "measurable" means that the maps sending a point to its color is Lebesgue Measurable). It is known for long that the measurable chromatic point of the plane is at least $5$ (Falconer 1981), so the interesting point here is that the lower bound gets better for large distances (i.e. for very negative curvature, if one prefers to fix $r=1$ and let $\kappa$ run over $(-\infty,0)$).

Note that under the Solovay axioms, the measurable hypothesis can be dropped since all sets are Lebesgue measurable (but then we don't have the full axiom of choice, and thus no DeBruin-Erdös theorem: we cannot deduce that under the Solovay axioms there is a finite unit-distance graph whose chromatic number is at least $6$).