What percent of the plane can be properly 5- and 6-colored?

Abstract

We present a tiling of more than 99.985698% of the Euclidean plane with six colors, reducing the previous record for uncovered fraction of the plane by about 12.8%. We also present a tiling of more than 95.99% of the plane with five colors. It is thus shown that any unit-distance graph of order at most 6992 and 24 in the plane can be properly 6-colored and 5-colored, respectively.