Every coseparable group may be free

拓扑Ramsey空间的每个子集在Solovay模型中都是Ramsey的

Every subset of a topological Ramsey space is Ramsey in the Solovay model

本文在Solovay模型下，无需超滤子和几乎归约，证明了拓扑Ramsey空间的每个子集都具有Ramsey性质。

Essentially disjoint families, conflict free colorings and Shelah's Revised GCH

Using Shelah's revised GCH theorem we prove that if mu<beth_omega <= lambda are cardinals, then every mu-almost disjoint subfamily B of [lambda]^{beth_omega} is essentially disjoint, i.e. for each b from B there is a subset f(b) of b of size < |b| such that the family {b-f(b) b in B} is disjoint. We also show that if mu<=kappa<=lambda, and kappa is infinite, and (x) every mu-almost disjoint subfamily of [lambda]^kappa is essentially disjoint, then (xx) every mu-almost disjoint family B of subsets of lambda with |b|>=kappa for all b from B has a conflict-free colorings with kappa colors. Puttin…

自由群作用的轨道等价与全弱混合

Orbit equivalence and total weak mixing of free group actions

该研究证明自由群的自由保测概率作用的轨道等价类含全弱混合作用，等价于特定代价的遍历可树化等价关系由自由全弱混合作用生成，解答了Miller与Tserunyan的问题。

Countably compact group topologies on arbitrarily large free Abelian groups

We prove that if there are $\mathfrak c$ incomparable selective ultrafilters then, for every infinite cardinal $κ$ such that $κ^ω=κ$, there exists a group topology on the free Abelian group of cardinality $κ$ without nontrivial convergent sequences and such that every finite power is countably compact. In particular, there are arbitrarily large countably compact groups. This answers a 1992 question of D. Dikranjan and D. Shakhmatov.

Hereditarily separable groups and monochromatic uniformization

We give a combinatorial equivalent to the existence of a non-free hereditarily separable group of cardinality aleph_1. This can be used, together with a known combinatorial equivalent of the existence of a non-free Whitehead group, to prove that it is consistent that every Whitehead group is free but not every hereditarily separable group is free. We also show that the fact that Z is a p.i.d. with infinitely many primes is essential for this result.

赋范空间的离散子群是自由的

Discrete subgroups of normed spaces are free

本文利用初等子模型理论与奇异紧致性定理，将Ancel等人关于赋范空间可数离散子群为自由Abel群的结论推广到任意基数的离散子群。

Equivariant maps to subshifts whose points have small stabilizers

Let $Γ$ be a countably infinite group. Given $k \in \mathbb{N}$, we use $\mathrm{Free}(k^Γ)$ to denote the free part of the Bernoulli shift action of $Γ$ on $k^Γ$. Seward and Tucker-Drob showed that there exists a free subshift $\mathcal{S} \subseteq \mathrm{Free}(2^Γ)$ such that every free Borel action of $Γ$ on a Polish space admits a Borel $Γ$-equivariant map to $\mathcal{S}$. Here we generalize this result as follows. Let $\mathcal{S}$ be a subshift of finite type (for example, $\mathcal{S}$ could be the set of all proper colorings of the Cayley graph of $Γ$ with some finite number of colo…