I came across a problem the other day that I learned was similar to the Hadwiger-Nelson Problem: If you have a 10 ft by 10 ft room, and you take the floor and color each point either red or blue, is there guaranteed to exist two points, exactly 1 foot apart, that are the same color? And the answer is obviously yes - take an equilateral triangle, for example.
But if we restrict the problem to only coloring the rational points of the floor, are we still guaranteed the same result? Obviously, you cannot have an equilateral triangle (in R^2) with strictly rational vertices. Restricting the problem to the irrationals or the integers yields an obvious yes, either through an equilateral triangle or a tesselation at each integer point, colored with diagonal lines being the same color. But I am quite stumped on the validity or solution to the problem with the rationals.