For any finite coloring of NxN there exists a square such that all corners are the same color.There are several proofs of this:
- This can be proven from the Hales-Jewitt theorem. (Advantage: automatically get out the full Gallai-Witt theorem with this approach.)
- This can be proven from the Gallai-Witt theorem trivially.
- This can be proven by using VDW theorem on the diagonal.
- This can be proven by using VDW theorem on the side.
- This can be proven by directly. It has a VDW flavor to it but can be shown to HS students who have not seen VDW theorem (I've done it).
For any finite coloring of NxN there exists an s &ge 2 and an s by s2 rectangle such that all corners are the same color.I have tried to prove this from scratch, from Poly vdw theorem, from Gallai-Witt, from ordinary Hales-Jewitt. Have not managed it.
- Is there a proof either from scratch, from poly vdw, or from Ordinary HJ? If so then please comment.
- Normally one asks for an elementary or (I prefer the phrase) purely combinatorial. Even so, I can't rephrase my question with this term since there is a purely combinatorial proof of Poly Hales-Jewitt. (See either Walters paper or the rough draft of the book on VDW stuff I am co-authoring with Andy Parrish pages 77-89.)