colours: If every point in the plane is covered with....

[malick]

This is from a Russian math contest.

If every point in the plane is covered with either red or blue. Given a rectangle A1A2A3A4, prove that it is possible to paint all its edges in red and blue colour.

I have no idea how to do this, does it involve the pigeon hole theorem?
Also does this apply if the plane is painted in 3,4,5, or 6 colors?

 

[stapel]

...prove that it is possible to paint all its edges in red and blue colour.

Do you mean "prove that it is possible to paint all its edges in either red or blue"?

I mean, if every point is red or blue, then certainly you can paint the edges with red and blue.

And are you trying to prove this that any rectangle can be so colored, or are you trying to prove that there exists a rectangle such that its edges can be of only one color?

Thank you.

Eliz.

 

[pka]

If every point in the plane is covered with either red or blue. Given a rectangle A1A2A3A4, prove that it is possible to paint all its edges in red and blue colour.

Surely this problem is misstated! Please check the wording.
There must be a definition of ‘color covering’.
Perhaps, the problem is about lattice points?
Does it say that “there exists a rectangle such that…”?
As stated, even with a definition of covering, the topology of the plane almost certainly makes the proposition false.

 

[malick]

Yes, I think that they meant to prove that it is possible to paint all its edges in either red or blue.

I mean, if every point is red or blue, then certainly you can paint the edges with red and blue.

 

[pka]

Without a lot more definition and/or clarification, the problem still makes no sense. There are problems about coloring the lattice points, points with integer coordinates. It may make some sense then- but not sure how.

 

[stapel]

I can find the following similar exercises:

Pigeonhole Problems (second page):

Every point of the plane is colored either red or blue. Show that there is a rectangle with vertices of the same color.

Cut-the-Knot - Pigeonhold Principle (#6):

Suppose each point of the plane is colored red or blue. Show that some rectangle has its vertices all the same color.

Proof Techniques (#5):

Suppose that each point in the plane is colored either red or blue.... Show that there is some rectangle all of whose vertices are the same color.

Hint: The trick with the pigeonhole principle is to figure out where to put your holes; once you get that, the rest falls into place nicely.... Consider a 3 × 9 grid of points (that is, 3 vertices vertically and 9 vertices horizontally). For any three vertices, there are 8 different possible color combinations for the vertices. We have 9 vertical sets of three vertices, so some two of them (by pigeonhole) must have the same coloring. Consider these two; each of them has (the same) two vertices either red or blue. Draw a rectangle using those vertices, and we are done.

Each of these refers to the vertices, rather than to the sides.

Please reply with the exact statement of the exercise. Thank you.

Eliz.

 

[pka]

Consider a 3 × 9 grid of points (that is, 3 vertices vertically and 9 vertices horizontally)

Those problems about coloring lattice points, are in fact well-known. But I must say that is much better solution than any I have ever seen.

 

[malick]

Sorry, it was the vertices. Copied the problem wrong.