Uniform density bound for six-colorings with a forbidden distance interval

Establish an upper bound \(\mu(C_1\cup\dots\cup C_6)\leq \mu_6^*<1\) for the fraction of the plane that can be correctly colored with six colors while forbidding all monochromatic distances in the interval \([1-\varepsilon,1+\varepsilon]\), for \(\varepsilon>0\), thereby implying \(\chi_{map}(\mathbb{R}^2)=7\).

Background

The conclusion observes that a uniform area-fraction bound for six-colorings with a forbidden distance interval would settle the unrestricted locally finite map-type coloring problem. The paper states that such a bound is not currently known and contrasts it with the weaker available result that a six-coloring can leave only a $1/6997$ fraction of the plane uncolored.

References

Note that if any upper bound $\mu(C_1 \cup \dots \cup C_6)\leq \mu*_6 <1$ on the fraction of the plane that can be correctly colored in 6 colors with forbidden interval $[1 - \varepsilon, 1+ \varepsilon]$, $\varepsilon>0$ were established, it would follow definitively that $\chi_{map}(\mathbb{R}2)=7.$ Unfortunately, we only know that there exists a 6-coloring in which the $1/6997$ part of the plane remains uncolored .

— On the chromatic number of the plane for map-type colorings  (2502.01958 - Sokolov et al., 4 Feb 2025) in Section Conclusion, paragraph following Question 2