Paint an infinite flat wall so that whenever two points are 1 meter apart, they are painted different colors. What's the least number of colors you need to do this?

The unit "1 meter" is of course not important. You have a wall which is infinite in both directions and flat, and you want to paint it using a few different colors, but not too many. Fix some unit distance and set yourself the goal of arranging that any two points that are that unit distance apart should be different colors. Can you paint this wall using only 2 different colors and achieve this goal? Or do you need 3? Or maybe 4? Or more? What is the least number of colors needed to acheive this. This is called the "chromatic number of the plane".