coloring combinatorics problem

Question

suppose i paint the plane two colors, white (W) and red (R). This means that every single point in the plane is either white or red.( think lattice ) A segment in the plane is called “good ” if both its ends have the same color. There are two types of good segments: if both ends of a good segment are white, we call this segment a “good white segment ”; otherwise we call it a “good red segment ”. 1. Prove that there is a good segment (white or red) of length 2015 miles. 2. Prove that there are infinitely many non-intersecting good segments of the same color, each of which has length 2015 miles. 3. Prove that there are infinitely many non-intersecting pairwise parallel good segments of the same color, each of which has a length of 2015 miles.

Answer 1

does this prove it in the general case ? pick a point O. With O as the center, draw a circle of radius d. There are just two possibilities:

The circle thus drawn contains a point of the same color as O. If that's the case we are finished OR -All points of the circle have a color different from the O. Then any chord of length d connects two points of the same color. Or does it mean integers in the plane are colored? the question is not specific enough is it ?

Answer 2

In my drawings, please treat BLUE points as WHITE. Blue just looks better in Geogebra than white :)

1. Prove that there is a good segment (white or red) of length 2015 miles.

Draw an equilateral triangle (1) with side length equal to 2015 miles. There are two colors and tri vertices. According to Dirichlet, at least two must be of the same color and one segment must be good, white or red.

2. Prove that there are infinitely many non-intersecting good segments of the same color, each of which has length 2015 miles.

Translate triangle (1) infinitely many times so that these triangles do not overalp. Such construction is definitely possible. Note that vertex colors change from one triangle to another.

As we proved in step one, there is a good segment in each triangle, either white or red. We have infinitely many triangles so we have infinitely many white or red good segments. If we had a finite number of good red segments and finite number of good white segments, the total number of good segments would be finite too.

But the total number of good segments is inifinite so we must have an infinite number of good segments, either of white or red color. And these segments obviously do not intersect.

3. Prove that there are infinitely many non-intersecting pairwise parallel good segments of the same color, each of which has a length of 2015 miles.

In step two we proved that we have an infinite number of good segments, either white or red. Because we used translation to construct all those triangles, some good segments of the same color are parallel to line AB, some to line BC and some to line CA. If we had finite number of good segments parallel to each of these three lines, the total number of good segments would be finite too.

But that number is infinite, as proved in step 2, so there must be an infinite number of good segments of the same color parallel with at least one of three possible directions (AB, BC or CA).