Monochromatic squares in a colored plane

Question

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.

Thank you

Answer 1

Molina, Oza, and Puttagunta, Sane bounds on Van der Waerden-type numbers, write,

The following is known: no matter how the lattice points of a coordinate plane are colored red and blue, there exists a square whose corners are the same color (a monochromatic square). In fact, using more than just two colors will still guarantee a monochromatic square (one whose vertices are the same color). So for all $c$ (the number of colors), there is a number $G(c)$ where all colorings with $c$ colors of the lattice points of a $G(c) \times G(c)$ grid will contain a monochromatic square. Unfortunately, the necessary number of points is unknown, but bounds are known. These bounds are enormous (roughly $G(c)\le2^{2^c}\times2^{2^{2^{2^{2c}}}}$).

The references they cite are

1. W. Gasarch, C. Kruskal, and A. Parrish. Van der Waerden’s theorem: Variants and applications. [But the Chapter that is supposed to be about "The Square Theorem" isn't there]

2. R. Graham, B. Rothchild, and J. Spencer. Ramsey Theory. Wiley, 1990.

3. B. Landmann and A. Robertson. Ramsey Theory over the integers. AMS, 2003.

Answer 2

The case for mono-color squares in a 2-colored grid has been solved:

• An infinite $13$-wide strip of grid can be colored to avoid mono squares
• a $14{\times}14$ grid can be colored to avoid mono squares
• a $14{\times}15$ grid must contain a mono-colored square

R Bacher and S Eliahou: Extremal binary matrices without constant 2-squares

A version of a $14{\times}14$ grid without a mono-colored square is shown, adapted from an image in the paper: