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A well known result in Ramsey theory is: If the set of positive integers is partitioned into a finite number of sets, then at least one of these sets will contain a solution to $x+y=z$

By "property" I mean arithmetic statements like "will contain a solution to $x+y=z$" and by "invariant under partition" I mean that whenever we partition the integers by a finite number of sets, at least one of the sets will have such property.

Is there any area of Ramsey theory dedicated to studying such types of problems? Where can I find references on more problems like that?

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  • $\begingroup$ It would be clearer to speak of partitioning $\mathbb Z$ 'into a finite number of subsets' instead of 'by a finite number of sets', if I understand your question correctly. $\endgroup$
    – Roland Bacher
    Commented Jan 17, 2022 at 16:42

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What you are looking for is called partition regularity, see the linked Wikipedia article for many examples of situations where this naturally occurs. The underlying set can be anything, it needn't be the set of integers $\mathbb Z$. Moreover, I would not consider this concept to be part of Ramsey theory; it is more widely used in combinatorics and combinatorial number theory.

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  • $\begingroup$ I guess that depends on one's definition of Ramsey theory. That same Wikipedia article you linked asserts "Ramsey theory is sometimes characterized as the study of which collections are partition regular"... $\endgroup$
    – Ale De Luca
    Commented Jan 18, 2022 at 9:07

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