Does Schur's Theorem Apply to Fractions in Ramsey Theory?

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SUMMARY

This discussion centers on the application of Schur's Theorem to fractions within the context of Ramsey Theory. The original poster has established basic bounds for Schur and Ramsey numbers and has demonstrated that an arithmetic sequence of fractions does not conform to Schur's Theorem. They hypothesize that Schur's Theorem may apply to fractions due to the relationship between Ramsey Theory and Schur's Theorem, and seek guidance on how to further explore this hypothesis.

PREREQUISITES
  • Understanding of Schur's Theorem
  • Familiarity with Ramsey Theory
  • Knowledge of arithmetic sequences
  • Basic skills in combinatorics
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  • Research fractional Ramsey Theory
  • Explore advanced combinatorial proofs related to Schur's Theorem
  • Study the implications of arithmetic sequences in Ramsey Theory
  • Review existing literature on the application of Schur's Theorem to non-integer values
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Mathematicians, researchers in combinatorics, and students studying Ramsey Theory who are interested in the intersection of Schur's Theorem and fractional applications.

huey910
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Hi all, I am currently doing research in Ramsey Theory and my research question is: does Schur's Theorem apply to fractions? Right now I am up to the point where I have proven several basic bounds of Schur and Ramsey numbers and have proven that an arithmetic sequence of fractions does not obey Schur's Theorem(I'm actually not sure whether my proof is correct, so I will post it up if anybody would like to check it for me).

Anyway, I searched online and found out that there is something called fractional ramsey theory and it led me to hypothesize that Schur's Theorem actually does apply to fractions(since Ramsey Theory and Schur's Theorem are related) but I'm not too sure where to start : can anybody help me? Thank you!
 
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and have proven that an arithmetic sequence of fractions does not obey Schur's Theorem(I'm actually not sure whether my proof is correct, so I will post it up if anybody would like to check it for me).

My skill in combinatorics is not so great that I could offer a good critique of your proof, but I would be very interested in seeing it.
 

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