More certainty in coloring the plane with a forbidden distance interval

Abstract

In the mysterious and colorful world of chromatic numbers, where there are a lot of unknown, there is an amazing thing. It turns out that for some intervals of forbidden distances on the plane, one can specify the exact value of the chromatic number $\chi$. Two sets of such intervals have been found, for $\chi=7$ and 9. We call them islands of certainty. Here we increase the size of these islands, and add three new ones with $\chi=8$, 12, 13. We also %formulate conjectures which predict conjecture islands for $\chi=14$, 15, 16. Are there islands of certainty for $\chi$=10 or 11? This is still a mystery. Roll up for the Mystery Tour.