The homeomorphism group of the universal Knaster continuum

We define a projective Fraiss\'e family whose limit approximates the universal Knaster continuum. The family is such that the group $\textrm{Aut}(\mathbb{K})$ of automorphisms of the Fraiss\'e limit is a dense subgroup of the group, $\textrm{Homeo}(K)$, of homeomorphisms of the universal Knaster continuum. We prove that both $\textrm{Aut}(\mathbb{K})$ and $\textrm{Homeo}(K)$ have universal minimal flow homeomorphic to the universal minimal flow of the free abelian group on countably many generators. The computation involves proving that both groups contain an open, normal subgroup which is extremely amenable.Comment: 27 pages, 1 figur

Fences, their endpoints, and projective Fra\"iss\'e theory

We introduce a new class of compact metrizable spaces, which we call fences, and its subclass of smooth fences. We isolate two families $\mathcal F, \mathcal F_0$ of Hasse diagrams of finite partial orders and show that smooth fences are exactly the spaces which are approximated by projective sequences from $\mathcal F_0$. We investigate the combinatorial properties of Hasse diagrams of finite partial orders and show that $\mathcal F, \mathcal F_0$ are projective Fra\"iss\'e families with a common projective Fra\"iss\'e limit. We study this limit and characterize the smooth fence obtained as its quotient, which we call a Fra\"iss\'e fence. We show that the Fra\"iss\'e fence is a highly homogeneous space which shares several features with the Lelek fan, and we examine the structure of its spaces of endpoints. Along the way we establish some new facts in projective Fra\"iss\'e theory.Comment: Version accepted for publication in the Transaction of the American Mathematical Societ

Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory

The goal of this present manuscript is to introduce the reader to the nonstandard method and to provide an overview of its most prominent applications in Ramsey theory and combinatorial number theory.Comment: 126 pages. Comments welcom