From abstract alpha-Ramsey theory to abstract ultra-Ramsey theory

Trujillo, Timothy

doi:10.48550/arxiv.1601.03831

Cited by 1 publication

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“…Note that a U -tree T contains no ⊑-maximal elements and that, for every s ∈ T , there is X ∈ [T ] such that s ⊑ X. The goal of this section is to prove the following Ramsey-theoretic statement about ultrafilter trees, recently proven by Trujillo in [108]: THEOREM 7.10. Suppose that U = U s : s ∈ N [<∞] is a sequence of non-principal ultrafilters on N, T is a U -tree on N, and X ⊆ N [∞] .…”

Section: Ultrafilter Treesmentioning

confidence: 99%

Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory

Nasso

Goldbring

Lupini

2019

Lecture Notes in Mathematics

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AcknowledgementsThe collaboration between the authors first began when they participated in an American Institute of Mathematics (AIM) Structured Quartet Research Ensemble (or SQuaRE) program together with Renling Jin, Steven Leth, and Karl Mahlburg. We thus want to thank AIM for all of their support during our three year participation in the SQuaRE program as well as their encouragement to organize a larger workshop on the subject. A preliminary version of this manuscript was distributed during that workshop and we want to thank the participants for their valuable comments. In particular, Steven Leth and Terence Tao gave us a tremendous amount of feedback and for that we want to give them an extra expression of gratitude. i ii INTRODUCTION iii and measurable sets in a dynamical system), with the extra feature that the dynamical system obtained perfectly reflects all the combinatorial properties of the set that one started with. The achievements of the nonstandard approach in this area include the work of Leth on arithmetic progressions in sparse sets, Jin's theorem on sumsets, as well as Jin's Freiman-type results on inverse problems for sumsets. More recently, these methods have also been used by Jin, Leth, Mahlburg, and the present authors to tackle a conjecture of Erdős concerning sums of infinite sets (the so-called B +C conjecture), leading to its eventual solution by Moreira, Richter, and Robertson.Nonstandard methods are also tightly connected with ultrafilter methods. This has been made precise and successfully applied in recent work of Di Nasso, where he observed that there is a perfect correspondence between ultrafilters and elements of the nonstandard universe up to a natural notion of equivalence. On the one hand, this allows one to manipulate ultrafilters as nonstandard points, and to use ultrafilter methods to prove the existence of certain combinatorial configurations in the nonstandard universe. One the other hand, this gives an intuitive and direct way to infer, from the existence of certain ultrafilter configurations, the existence of corresponding standard combinatorial configurations via the fundamental principle of transfer in the nonstandard method. This perspective has successfully been applied by Di Nasso, Luperi Baglini, and co-authors to the study of partition regularity problems for Diophantine equations over the integers, providing in particular a far-reaching generalization of the classical theorem of Rado on partition regularity of systems of linear equations. Unlike Rado's theorem, this recent generalization also includes equations that are not linear.Finally, it is worth mentioning that many other results in combinatorics can be seen, directly or indirectly, as applications of the nonstandard method. For instance, the groundbreaking work of Hrushovski and Breuillard-Green-Tao on approximate groups, although not originally presented in this way, admit a natural nonstandard treatment. The same applies to the work of Bergelson and Tao on recurrence in quasirandom groups.The goal of t...

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Section: Ultrafilter Treesmentioning

confidence: 99%