On the optimality of coloring with a lattice

For z₁, z₂, z₃ ∈ Z², the tristance d₃(z₁, z₂, z₃) is a generalization of the L₁-distance on Z² to a quality that reflects the relative dispersion of three points rather than two. We prove that at least 3k² colors are required to color the points of Z², such that the tristance between any three distinct points, colored with the same color, is at least 4k. We also prove that 3k²+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 3k² colors and conjecture that these are the only colorings with 3k² colors.