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?



References for properties which are invariant under partition of $\mathbb{Z}$ by a finite number of sets
What you are looking for is called [partition regularity](https://en.wikipedia.org/wiki/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.


