Abstract
Ramsey theory initially focused on the study of combinatorial properties that are “abundant” in the sense that can always be found in one of the pieces of any finite partition of a given structure. Such a notion can be strengthened by considering combinatorial configuration that can be found in any set that is “large” in a more generous quantitative sense.
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Notes
- 1.
At this point, we may note that G satisfies the conclusion of the proposition except that it is not an interval but instead a Følner approximation for \(\mathbb {Z}\). While this would suffice for our purposes in Sect. 12.5, we wanted to avoid having to introduce the theory of Følner approximations and instead opted to work a bit harder to obtain the above cleaner statement.
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Nasso, M.D., Goldbring, I., Lupini, M. (2019). Densities and Structural Properties. In: Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory. Lecture Notes in Mathematics, vol 2239. Springer, Cham. https://doi.org/10.1007/978-3-030-17956-4_10
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