The following is known: For all $c$, for all $c$-colorings of $N\times N$ there exists a $d \times d^2$ rectangle ($d \ge 2$) such that all four corners are the same color.

The proof uses the Poly-Hales-Jewitt theorem.

I seek an easier proof. Okay to use Van der Waerden's theorem (VDW) or poly VDW or Gallai-Witt theorem.

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    $\begingroup$ Did you know that you can format the mathematics in your question using latex and other tricks? It'll make the question easier to read. Also it's better to spell things out more explicitly. Using abbreviations such as "Elem" makes the question look less professional. Finally, providing a link for VDW would make it easier for people to grasp your question. $\endgroup$ Feb 26, 2011 at 22:02
  • $\begingroup$ It seems to me that this question would be more likely to get an answer on MathOverflow. $\endgroup$ Feb 27, 2011 at 12:21


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