اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدSuperfilters, Ramsey theory, and van der Waerdens Theorem
357
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Superfilters are generalized ultrafilters, which capture the underlying concept in Ramsey theoretic theorems such as van der Waerdens Theorem. We establish several properties of superfilters, which generalize both Ramseys Theorem and its variant for ultrafilters on the natural numbers. We use them to confirm a conjecture of Kov{c}inac and Di Maio, which is a generalization of a Ramsey theoretic result of Scheepers, concerning selections from open covers. Following Bergelson and Hindmans 1989 Theorem, we present a new simultaneous generalization of the theorems of Ramsey, van der Waerden, Schur, Folkman-Rado-Sanders, Rado, and others, where the colored sets can be much smaller than the full set of natural numbers.
قيم البحث
اقرأ أيضاً
We prove a canonical polynomial Van der Waerdens Theorem. More precisely, we show the following. Let ${p_1(x),ldots,p_k(x)}$ be a set of polynomials such that $p_i(x)in mathbb{Z}[x]$ and $p_i(0)=0$, for every $iin {1,ldots,k}$. Then, in any colouring
of $mathbb{Z}$, there exist $a,din mathbb{Z}$ such that ${a+p_1(d),ldots,a+p_{k}(d)}$ forms either a monochromatic or a rainbow set.
We explore the properties of non-piecewise syndetic sets with positive upper density, which we call discordant, in countable amenable (semi)groups. Sets of this kind are involved in many questions of Ramsey theory and manifest the difference in compl
exity between the classical van der Waerdens theorem and Szemeredis theorem. We generalize and unify old constructions and obtain new results about these historically interesting sets. Here is a small sample of our results. $bullet$ We connect discordant sets to recurrence in dynamical systems, and in this setting we exhibit an intimate analogy between discordant sets and nowhere dense sets having positive measure. $bullet$ We introduce a wide-ranging generalization of the squarefree numbers, producing many examples of discordant sets in $mathbb{Z}$, $mathbb{Z}^d$, and the Heisenberg group. We develop a unified method to compute densities of these discordant sets. $bullet$ We show that, for any countable abelian group $G$, any F{o}lner sequence $Phi$ in $G$, and any $c in (0, 1)$, there exists a discordant set $A subseteq G$ with $d_Phi(A) = c$. Here $d_Phi$ denotes density along $Phi$. Along the way, we draw from various corners of mathematics, including classical Ramsey theory, ergodic theory, number theory, and topological and symbolic dynamics.
In this paper we establish a new connection between central sets and the strong coincidence conjecture for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from topological
dynamics, constitute a special class of subsets of $ ats$ possessing strong combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. We give an equivalent reformulation of the strong coincidence condition in terms of central sets and minimal idempotent ultrafilters in the Stone-v{C}ech compactification $beta ats .$ This provides a new arithmetical approach to an outstanding conjecture in tiling theory, the Pisot substitution conjecture. The results in this paper rely on interactions between different areas of mathematics, some of which had not previously been directly linked: They include the general theory of combinatorics on words, abstract numeration systems, tilings, topological dynamics and the algebraic/topological properties of Stone-v{C}ech compactification of $ ats.$
Inspired by Ramseys theorem for pairs, Rival and Sands proved what we refer to as an inside/outside Ramsey theorem: every infinite graph $G$ contains an infinite subset $H$ such that every vertex of $G$ is adjacent to precisely none, one, or infinite
ly many vertices of $H$. We analyze the Rival-Sands theorem from the perspective of reverse mathematics and the Weihrauch degrees. In reverse mathematics, we find that the Rival-Sands theorem is equivalent to arithmetical comprehension and hence is stronger than Ramseys theorem for pairs. We also identify a weak form of the Rival-Sands theorem that is equivalent to Ramseys theorem for pairs. We turn to the Weihrauch degrees to give a finer analysis of the Rival-Sands theorems computational strength. We find that the Rival-Sands theorem is Weihrauch equivalent to the double jump of weak K{o}nigs lemma. We believe that the Rival-Sands theorem is the first natural theorem shown to exhibit exactly this strength. Furthermore, by combining our result with a result of Brattka and Rakotoniaina, we obtain that solving one instance of the Rival-Sands theorem exactly corresponds to simultaneously solving countably many instances of Ramseys theorem for pairs. Finally, we show that the uniform computational strength of the weak Rival-Sands theorem is weaker than that of Ramseys theorem for pairs by showing that a number of well-known consequences of Ramseys theorem for pairs do not Weihrauch reduce to the weak Rival-Sands theorem. We also address an apparent gap in the literature concerning the relationship between Weihrauch degrees corresponding to the ascending/descending sequence principle and the infinite pigeonhole principle.
Let ${mathfrak C}$ be a monster model of an arbitrary theory $T$, $bar alpha$ any tuple of bounded length of elements of ${mathfrak C}$, and $bar c$ an enumeration of all elements of ${mathfrak C}$. By $S_{bar alpha}({mathfrak C})$ denote the compact
space of all complete types over ${mathfrak C}$ extending $tp(bar alpha/emptyset)$, and $S_{bar c}({mathfrak C})$ is defined analogously. Then $S_{bar alpha}({mathfrak C})$ and $S_{bar c}({mathfrak C})$ are naturally $Aut({mathfrak C})$-flows. We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of ${mathfrak C}$), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend on the choice of the monster model ${mathfrak C}$; thus, we say that they are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows $S_{bar alpha}({mathfrak C})$ and $S_{bar c}({mathfrak C})$. We give an example of a NIP theory in which the minimal left ideals are of unbounded size. We show that in each of these two cases, boundedness of a minimal left ideal is an absolute property (i.e. it does not depend on the choice of ${mathfrak C}$) and that whenever such an ideal is bounded, then its isomorphism type is also absolute. Assuming NIP, we give characterizations of when a minimal left ideal of the Ellis semigroup of $S_{bar c}({mathfrak C})$ is bounded. Then we adapt a proof of Chernikov and Simon to show that whenever such an ideal is bounded, the natural epimorphism (described by Krupinski, Pillay and Rzepecki) from the Ellis group of the flow $S_{bar c}({mathfrak C})$ to the Kim-Pillay Galois group $Gal_{KP}(T)$ is an isomorphism (in particular, $T$ is G-compact). We provide some counter-examples for $S_{bar alpha}({mathfrak C})$ in place of $S_{bar c}({mathfrak C})$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
