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Albert Einstein

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Each point in a plane is colored red or blue (the red and blue are very weak, practically white). If you find a rectangle which has all its vertices on points of the same color you can fill it with black. Prove that it is possible to construct such black rectangles in the plane so that they together with the remaining white area, to any desired degree, look like a black and white picture of any resolution of Albert Einstein, like the one below.

In other words, for any "pixel" you choose, no matter how small it is, show that it will be possible to construct black rectangles so that the pixel gets the right ratio of black and white to any desired degree of accuracy.

Solution