Applications of Ramsey's Theorem

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Ramsey's Theorem has diverse applications across various fields of mathematics, including combinatorics, graph theory, and number theory. It aids in understanding the structure of large sets and the existence of certain configurations within them. Notable applications include the study of coloring problems in graphs and the analysis of combinatorial structures. Additionally, it plays a role in computer science, particularly in algorithms and complexity theory. Overall, Ramsey's Theorem is fundamental in exploring the inherent order within chaos in mathematical contexts.
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Hey, Guys!

I wonder what applications of Ramsey's famous theorem there exist apart from such cases as:
1) constructions of functions in the domain of natural numbers to check some properties of sequences of real numbers.
2) propositions about points in convex postition (Erdos-Szekeres Problem)

I don't ask for detailed descriptions, although they are also welcome.
You can simply give a short sketch of applications in different fields of maths or give a link.

Thanks in advance! :smile:
 
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