Prove that the distance between a black and a white dot is one

If you click on the link you find a picture which has black dots on a white background. This suggests that we color every point on the plane either white or black, making sure to use infinitely many of each color. With these constraints it is indeed the case that there must be a black point and a white point at unit distance. (In fact, "infinite" can be weakened to nonempty.)

Hint: starting with a black point, we're done unless the entire unit circle around that point is black. Now repeat that argument for each point on that unit circle: we've already generated a sizable swathe of the plane colored totally black...

The statement is in general false: for example, if the black dots are all and only on the locus $x=0$, while the white ones are all and only on the locus $x=2$, then the assertion is false (we are using the Euclidean distance).

Probably there are some more hypothesis in the background OR the aim of the question is just to analyze the way the candidate faces a given mathematical problem, reaching conclusions which can be in contrast with the thesis of any given question.

Another counterexample. Take the real line with every integer coloured white and all other points coloured black.

On the other hand if every point in the whole plane is coloured white or black, with infinite numbers of each, and we try to build a counterexample, the following happens. There is at least one white point $P$. To avoid black points at distance $1$ we colour the circumference of the unit circle centred at $P$ white too. Then every point in the circle is unit distance from some point on the circumference, to so the interior of the circle has to be white. Eventually we conclude that the whole plane is white, which contradicts the existence of a black point.