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

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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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