ترغب بنشر مسار تعليمي؟ اضغط هنا

Ramsey degrees of ultrafilters, pseudointersection numbers, and the tools of topological Ramsey spaces

85   0   0.0 ( 0 )
 نشر من قبل Natasha Dobrinen
 تاريخ النشر 2019
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

This paper investigates properties of $sigma$-closed forcings which generate ultrafilters satisfying weak partition relations. The Ramsey degree of an ultrafilter $mathcal{U}$ for $n$-tuples, denoted $t(mathcal{U},n)$, is the smallest number $t$ such that given any $lge 2$ and coloring $c:[omega]^nrightarrow l$, there is a member $Xinmathcal{U}$ such that the restriction of $c$ to $[X]^n$ has no more than $t$ colors. Many well-known $sigma$-closed forcings are known to generate ultrafilters with finite Ramsey degrees, but finding the precise degrees can sometimes prove elusive or quite involved, at best. In this paper, we utilize methods of topological Ramsey spaces to calculate Ramsey degrees of several classes of ultrafilters generated by $sigma$-closed forcings. These include a hierarchy of forcings due to Laflamme which generate weakly Ramsey and weaker rapid p-points, forcings of Baumgartner and Taylor and of Blass and generalizations, and the collection of non-p-points generated by the forcings $mathcal{P}(omega^k)/mathrm{Fin}^{otimes k}$. We provide a general approach to calculating the Ramsey degrees of these ultrafilters, obtaining new results as well as streamlined proofs of previously known results. In the second half of the paper, we calculate pseudointersection and tower numbers for these $sigma$-closed forcings and their relationships with the classical pseudointersection number $mathfrak{p}$.



قيم البحث

اقرأ أيضاً

We study the topological version of the partition calculus in the setting of countable ordinals. Let $alpha$ and $beta$ be ordinals and let $k$ be a positive integer. We write $betato_{top}(alpha,k)^2$ to mean that, for every red-blue coloring of the collection of 2-sized subsets of $beta$, there is either a red-homogeneous set homeomorphic to $alpha$ or a blue-homogeneous set of size $k$. The least such $beta$ is the topological Ramsey number $R^{top}(alpha,k)$. We prove a topological version of the ErdH{o}s-Milner theorem, namely that $R^{top}(alpha,k)$ is countable whenever $alpha$ is countable. More precisely, we prove that $R^{top}(omega^{omega^beta},k+1)leqomega^{omega^{betacdot k}}$ for all countable ordinals $beta$ and finite $k$. Our proof is modeled on a new easy proof of a weak version of the ErdH{o}s-Milner theorem that may be of independent interest. We also provide more careful upper bounds for certain small values of $alpha$, proving among other results that $R^{top}(omega+1,k+1)=omega^k+1$, $R^{top}(alpha,k)< omega^omega$ whenever $alpha<omega^2$, $R^{top}(omega^2,k)leqomega^omega$ and $R^{top}(omega^2+1,k+2)leqomega^{omegacdot k}+1$ for all finite $k$. Our computations use a variety of techniques, including a topological pigeonhole principle for ordinals, considerations of a tree ordering based on the Cantor normal form of ordinals, and some ultrafilter arguments.
We define a collection of topological Ramsey spaces consisting of equivalence relations on $omega$ with the property that the minimal representatives of the equivalence classes alternate according to a fixed partition of $omega$. To prove the associa ted pigeonhole principles, we make use of the left-variable Hales-Jewett theorem and its extension to an infinite alphabet. We also show how to transfer the corresponding infinite-dimensional Ramsey results to equivalence relations on countable limit ordinals (up to a necessary restriction on the set of minimal representatives of the equivalence classes) in order to obtain a dual Ramsey theorem for such ordinals.
265 - L. Nguyen Van The 2009
In 2003, Kechris, Pestov and Todorcevic showed that the structure of certain separable metric spaces - called ultrahomogeneous - is closely related to the combinatorial behavior of the class of their finite metric spaces. The purpose of the present p aper is to explore the different aspects of this connection.
In this paper, we consider a variant of Ramsey numbers which we call complementary Ramsey numbers $bar{R}(m,t,s)$. We first establish their connections to pairs of Ramsey $(s,t)$-graphs. Using the classification of Ramsey $(s,t)$-graphs for small $s, t$, we determine the complementary Ramsey numbers $bar{R}(m,t,s)$ for $(s,t)=(4,4)$ and $(3,6)$.
120 - L. Nguyen Van The 2007
Given a countable set S of positive reals, we study finite-dimensional Ramsey-theoretic properties of the countable ultrametric Urysohn space with distances in S.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا