On avoding r-repetitions in R2

We say that a coloring of R2 is line k-nonrepetitive if we will never encounter the same sequence of colors k times in a row when going along any straight line with steps of length 1 unit. This notion is closely related to square- and cube-free words, introduced by Thue.We show that there exists a line 40-nonrepetitive coloring of the plane with only 2 colors. Moreover, we generalize this result for higher dimensions and more general patterns. It complements recent results concerning line 2-nonrepetitive colorings of the plane.