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.
More elementary proof of coloring theorem for d x d^2 rectangles
co.combinatorics;proofs;ramsey theory