Survey on the Tukey theory of ultrafilters

This article surveys results regarding the Tukey theory of ultrafilters on countable base sets. The driving forces for this investigation are Isbell's Problem and the question of how closely related the Rudin-Keisler and Tukey reducibilities are. We review work on the possible structures of cofinal types and conditions which guarantee that an ultrafilter is below the Tukey maximum. The known canonical forms for cofinal maps on ultrafilters are reviewed, as well as their applications to finding which structures embed into the Tukey types of ultrafilters. With the addition of some Ramsey theory, fine analyses of the structures at the bottom of the Tukey hierarchy are made.Comment: 25 page

Linear extensions of partial orders and Reverse Mathematics

We introduce the notion of \tau-like partial order, where \tau is one of the linear order types \omega, \omega*, \omega+\omega*, and \zeta. For example, being \omega-like means that every element has finitely many predecessors, while being \zeta-like means that every interval is finite. We consider statements of the form "any \tau-like partial order has a \tau-like linear extension" and "any \tau-like partial order is embeddable into \tau" (when \tau\ is \zeta\ this result appears to be new). Working in the framework of reverse mathematics, we show that these statements are equivalent either to B\Sigma^0_2 or to ACA_0 over the usual base system RCA_0.Comment: 8 pages, minor changes suggested by referee. To appear in MLQ - Mathematical Logic Quarterl

Ramsey Properties of Countably Infinite Partial Orderings

A partial ordering ℙ is chain-Ramsey if, for every natural number n and every coloring of the n-element chains from ℙ in finitely many colors, there is a monochromatic subordering ℚ isomorphic to ℙ. Chain-Ramsey partial orderings stratify naturally into levels. We show that a countably infinite partial ordering with finite levels is chain-Ramsey if and only if it is biembeddable with one of a canonical collection of examples constructed from certain edge-Ramsey families of finite bipartite graphs. A similar analysis applies to a large class of countably infinite partial orderings with infinite levels

Laver and set theory

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

Ramsey precompact expansions of homogeneous directed graphs

In 2005, Kechris, Pestov and Todorcevic provided a powerful tool to compute an invariant of topological groups known as the universal minimal flow, immediately leading to an explicit representation of this invariant in many concrete cases. More recently, the framework was generalized allowing for further applications, and the purpose of this paper is to apply these new methods in the context of homogeneous directed graphs. In this paper, we show that the age of any homogeneous directed graph allows a Ramsey precompact expansion. Moreover, we verify the relative expansion properties and consequently describe the respective universal minimal flows