1

It’s an extension of Schur’ Theorem. I need to proof that there exist $N$ such that for any coloring of first $N$ natural numbers in three colors there will be three one colored numbers $x, y, z$ such that $x + y = 2z$.

Edit. I came up with this equation when I was trying to prof that W(3, 3) is finite (i.e. there exist n such that for every 3-coloring of first n natural numbers one can always find an ariphmetic progression of three numbers x < z < y (equivalent to y - z = z - x <=> x + y = 2z)).

Daniil
  • 93
  • 2
    You've been here a while; you should already know that questions that merely contain a problem statement are not acceptable. Also, you misspelled Schur. – Misha Lavrov Dec 17 '23 at 06:59
  • Well I know Schur’ theorem proof but simply reproduce it is a nonsense since there are a lot of sources where one can read about it. Considering this problem — I was trying to prove that W(3, 3) is finite. I wanted to reduce it to equation x + y = 2z, but I don’t have a clue how to reference to Schur. Also you don’t sound friendly enough. – Daniil Dec 17 '23 at 07:27
  • From How to ask a good question: "Avoid "no clue" questions. Too many questions begin or end with "I don't even know how to begin with this problem". While this may be true (you may genuinely have no idea how to approach the problem), it is still not a valid reason to limit your post to the statement of the problem without any mention of your own thoughts. Such questions will most of the time be rejected by the community, which represents a significant waste of time - including for yourself - ..." – jjagmath Dec 17 '23 at 08:10
  • Well I disagree that writing this post is waste of anyone's time. Although your comment might be. I hope someone will answer my post - if not I tried at least. As for reasons - not having a clue what to do on a certain step is a valid reason not to write all attempts that failed. – Daniil Dec 17 '23 at 08:26
  • If you had tried writing a program to find the least such $N$, you might have found $N=14$. – quasi Dec 17 '23 at 11:35
  • I was just sharing the policies of the forum with you. If you don't agree with them, that's ok. But if you continue to post here without following the rules, you should expect similar critics to your questions. – jjagmath Dec 17 '23 at 12:14

0 Answers0