 Solution

Alternating suitable half-open vertical strips of width $\sqrt{3}/2$ gives a two-colouring in which the horizontal projections forced by a unit equilateral triangle cannot all occupy strips of one parity; boundary strips are assigned consistently. For three colours, embed the Moser spindle, a finite unit-distance graph of chromatic number four, in the plane. Any three-colouring therefore has a monochromatic unit edge.