On the optimality of coloring with a lattice

For z/sub 1/, z/sub 2/, z/sub 3/ /spl isin/ Z/sup 2/, the tristance d/sub 3/(z/sub 1/, z/sub 2/, z/sub 3/) is a generalization of the L/sub 1/-distance on Z/sup 2/ to a quality that reflects the relative dispersion of three points rather than two. We prove that at least 3k/sup 2/ colors are required to color the points of Z/sup 2/, such that the tristance between any three distinct points, colored with the same color, is at least 4k. We also prove that 3k/sup 2/+3k+1 colors are required if the tristance is at least 4k+2. For the first case we show an infinite family of colorings with 3k/sup 2/ colors and conjecture that these are the only colorings with 3k/sup 2/ colors.