Color every point in the real plane using the colors blue,yellow only. It can be shown that there exists a rectangle that has all vertices with the same color. Is it possible to show that there exists a square that has all vertices with the same color ? If it is not possible, please give me an example of a coloring of the real plane that does not have monochromatic squares.
To give the viewers an idea about other similar results (which they might find useful), for any coloring (2 colors) of the real plane:
1) There exists three collinear points having the same color, such that one of the points is the midpoint of the line segment that joins the other two.
2)For any two angles $\theta,\phi$ there exists a monochromatic triangle that has angles $\theta,\phi,180-(\theta+\phi)$
3)For any angle $\theta$, there exists a monochromatic parallelogram with angle $\theta$
Now its natural to ask if there are any monochromatic squares.