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

In summary, the person is researching Ramsey Theory and wondering if Schur's Theorem applies to fractions. They have proven basic bounds of Schur and have doubts about their proof. They have discovered fractional Ramsey Theory and have a hypothesis that Schur's Theorem applies to fractions. They are seeking help and are willing to share their proof for feedback.
  • #1
huey910
36
0
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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  • #2
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.
 

1. What is Ramsey Theory?

Ramsey Theory is a branch of mathematics that studies the emergence of order in seemingly random structures. It is named after the mathematician Frank P. Ramsey and is a fundamental field in combinatorics and graph theory.

2. What is the main goal of research in Ramsey Theory?

The main goal of research in Ramsey Theory is to find patterns and regularities in complex structures by identifying and characterizing the conditions under which order must emerge. This involves studying the existence and properties of certain substructures within larger, seemingly chaotic structures.

3. What are some real-world applications of Ramsey Theory?

Ramsey Theory has applications in various fields, such as computer science, economics, and social sciences. It has been used to solve problems in scheduling, network design, and even in understanding cooperation and conflict in human societies.

4. What are some current research topics in Ramsey Theory?

Some current research topics in Ramsey Theory include the study of Ramsey numbers, which determine the minimum size of a structure that guarantees a certain level of order, and the investigation of Ramsey properties in infinite structures, such as infinite graphs and topological spaces.

5. How does Ramsey Theory relate to other areas of mathematics?

Ramsey Theory has connections to a wide range of mathematical fields, including combinatorics, graph theory, topology, logic, and number theory. It also has applications in theoretical computer science and has been used to prove results in other areas of mathematics, such as geometry and group theory.

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