Knowing that every point in a plane is either black or white ...

Question

Knowing that every point in a plane is either black or white, I need to prove that:

• a segment $AB$ of length $7$ with its vertices $A$ and $B$ of the same colour always exists.

• a coloration of the plane always exists by which in every vertical segment $AB$ of length $7$ the vertices $A$ e $B$ never have the same colour.

• at least one coloration of the plane exists by which no open part of a rectangle has the same colour.

Honestly, I've never dealt with problems like this. My question is: since a plane is made up of an infinite number of points, isn't it obvious that the first and the second requests are true?

Moreover, I have no idea what “open part of a rectangle” (textually) stands for.

Answer 1

For the first one, fix a point $P,$ and consider all points at a distance of $7$ from $P.$ If one of them is the same color as $P,$ we're done. If not, then the set of all such points forms a circle of points about $P$ of radius $7,$ each of which is the same color. Can you take it from there?

For the second, try coloring all points $(x,y)$ with $0\le y<7$ black. Can you see how to take it from there and color all the other points?

For the third, I'm not sure what "open part of a rectangle" could mean, but I suspect that it means the interior of the rectangle, and that we're to find a coloration such that no rectangle has an interior that is all one color. If that is correct, consider the coloration of the plane by making $(x,y)$ white when $x$ is a rational number, and black when $x$ is irrational.

Unfortunately, I can't be sure I've understood the third part of your question correctly. I note, however, that you say "$A$ e $B$" (instead of "$A$ and $B$") in your second question, which makes me suspect that English is not your primary language, and that the problem was originally posed in another language. I suggest that you consider editing your post to include the problem in its original language, as well, and add the [translation-request] tag. Hopefully, one of our users who is more conversant in both languages will be able to translate the third part of your question more accurately and clearly.

Answer 2

For part 1, consider an equilateral triangle with side length $7$. It is easy to see that at least two of this triangle's three edges must have same-colour endpoints.

A colouring of the plane satisfying the conditions in part 2 colours exactly those vertices $(x,y)$ with $y\bmod14\in[0,7)$ black, and the rest white. The open/closedness of the endpoints of $[0,7)$ is significant here.

Similarly, a colouring satisfying the third question's conditions has $(x,y)$ black iff $x,y$ are both rational. Every non-empty rectangle (the open part is the interior) will have a horizontal or vertical line segment, and thus infinitely many white and black points.