C. Laflamme, L. Nguyen Van The, M. Pouzet, N. Sauer
We investigate infinite versions of vector and affine space partition results, and thus obtain examples and a counterexample for a partition problem for relational structures. In particular we provide two (related) examples of an age indivisible relational structure which is not weakly indivisible.
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Indivisibility and uniform computational strength
该研究聚焦有理数线性序与可数等价类等价关系两类结构的不可分性问题，运用Weihrauch复杂度分析方法，明确了对应问题的Weihrauch度与多个基准的分离关系，推进了拉姆齐定理变体相关的反推数学研究。
Prompted by a recent question of G. Hjorth as to whether a bounded Urysohn space is indivisible, that is to say has the property that any partition into finitely many pieces has one piece which contains an isometric copy of the space, we answer this question and more generally investigate partitions of countable metric spaces. We show that an indivisible metric space must be totally Cantor disconnected, which implies in particular that every Urysohn space U_V with V bounded or not but dense in some initial segment of R+, is divisible. On the other hand we also show that one can remove "large" …
Effectiveness and strong graph indivisibility
本研究用可计算性理论和反数学工具分析Cameron强图不可分性定理的强度，证明其可计算表示下有效、部分结果支持全定理在$ω$-模型$\u210e\boxed{REC}$中成立，且原证明本质用$\u210e\boxed{I}Σ^0_2$归纳方案。
A structure $\mathcal{M}$ in a first-order language $\mathcal{L}$ is \emph{indivisible} if for every coloring of $M$ in two colors, there is a monochromatic $\mathcal{M}^{\prime} \subseteq \mathcal{M}$ such that $\mathcal{M}^{\prime}\cong\mathcal{M}$. Additionally, we say that $\mathcal{M}$ is symmetrically indivisible if $\mathcal{M}^{\prime}$ can be chosen to be \emph{symmetrically embedded} in $\mathcal{M}$ (that is, every automorphism of $\mathcal{M}^{\prime}$ can be extended to an automorphism of $\mathcal{M}$). In the following paper we give a general method for constructing new symmetri…
Dedekind-finite cardinals having countable partitions
本文研究没有可数子集但在两种满射意义上非“surjectively Dedekind finite”的集合可能具有的结构。
We study the Baire class one countable colorings, i.e., the countable partitions into $F_σ$ sets. Such a partition gives a covering of the diagonal into countably many $F_σ$ squares. This leads to the study of countable unions of $F_σ$ rectangles. We give a Hurewicz-like dichotomy for such countable unions.
The strongest type of coloring of pairs of countable ordinals, gotten by Todorcevic from a strongly Luzin set, is shown to be equivalent to the existence of a nonmeager set of reals of size $\aleph_1$. In the other direction, it is shown that the existence of both a strongly Luzin set and a coherent Souslin tree is compatible with the existence of a countable partition of pairs of countable ordinals such that no coloring is strong over it. This clarifies the interaction between a gallery of coloring assertions going back to Luzin and Sierpinski a hundred years ago.
On closed Ramsey numbers of small countable ordinals
本文研究了可数序数对的闭合划分关系，证明了闭合Ramsey数的改进上界和下界，同时给出了拓扑划分序数的必要条件。
We analyse partitions of products with two ordered factors in two classes where both factors are countable or well-ordered and at least one of them is countable. This relates the partition properties of these products to cardinal characteristics of the continuum. We build on work by Erdős, Garti, Jones, Orr, Rado, Shelah and Szemerédi. In particular, we show that a theorem of Jones extends from the natural numbers to the rational ones but consistently extends only to three further equimorphism classes of countable orderings. This is made possible by applying a thirteen-year old theorem of Orr …
We define a collection of topological Ramsey spaces consisting of equivalence relations on $ω$ with the property that the minimal representatives of the equivalence classes alternate according to a fixed partition of $ω$. To prove the associated 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) i…
We study the finite dimensional partition properties of the countable homogeneous dense local order. Some of our results use ideas borrowed from the partition calculus of the rationals and are obtained thanks to a strengthening of Milliken's theorem on trees.
Infinite-Exponent Partition Relations on Higher Analogues of the Real Line
本文研究了形如⟨^α2,<_lex⟩的线性序的无限指数划分关系，在ZF中无选择公理下，给出了实线设定的推广结果，并对⟨^α2,<_lex⟩→(τ)^τ关系进行了完整分类。
This article highlights historical achievements in the partition theory of countable homogeneous relational structures, and presents recent work, current trends, and open problems. Exciting recent developments include new methods involving logic, topological Ramsey spaces, and category theory. The paper concentrates on big Ramsey degrees, presenting their essential structure where known and outlining areas for further development. Cognate areas, including infinite dimensional Ramsey theory of homogeneous structures and partition theory of uncountable structures, are also discussed.
Given a countable graph, we say a set $A$ of its vertices is \emph{universal} if it contains every countable graph as an induced subgraph, and $A$ is \emph{weakly universal} if it contains every finite graph as an induced subgraph. We show that, for almost every graph on $\mathbb N$, $(1)$ every set of positive upper density is universal, and $(2)$ every set with divergent reciprocal sums is weakly universal. We show that the second result is sharp (i.e., a random graph on $\mathbb N$ will almost surely contain non-universal sets with divergent reciprocal sums) and, more generally, that neithe…
The \textit{nowhere dense ideal} $\mathcal{NC}$ is introduced. It is a coanalytic ideal of $ω\timesω$ whose defining characteristic is that the sets of the form $X\times Y$, where $X,Y$ are infinite subsets of $ω$, are dense in the quotient $\mathcal{P}(ω\timesω)\slash\mathcal{NC}$. This quotient has countable partitions and consistently has partitions of size $ω_1$ while $\mathfrak{a}>ω_1$. This represents a huge contrast with other definable quotients. Other combinatorial features of this ideal are presented, as well as some results on a family of similar, higher dimensional ideals.
We investigate the existence of metric spaces which, for any coloring with a fixed number of colors, contain monochromatic isomorphic copies of a fixed starting space K. In the main theorem we construct such a space of size \(2^{\aleph_0}\) for colorings with \(\aleph_0\) colors and any metric space \(K\) of size \(\aleph_0\). We also give a slightly weaker theorem for countable ultrametric \(K\) where, however, the resulting space has size~\(\aleph_1\).
All spaces of countable spread can be small
本文证明了在弱可 compact 卡尔文条件下，可数扩张空间的拓扑性质与基数关系
A countable-support symmetric iteration separating PP from AC
本文通过可数支持对称迭代构造一个满足ZF+DC+PP+AC_{wo}+¬AC的对称模型，利用轨道对称化包和局部分裂原理实现PP与AC的分离。
We investigate negative square-brackets partition relations at successors of singular cardinals of countable cofinality. Along the way we prove some club-guessing results.
We study the topological version of the partition calculus in the setting of countable ordinals. Let $α$ and $β$ be ordinals and let $k$ be a positive integer. We write $β\to_{top}(α,k)^2$ to mean that, for every red-blue coloring of the collection of 2-sized subsets of $β$, there is either a red-homogeneous set homeomorphic to $α$ or a blue-homogeneous set of size $k$. The least such $β$ is the topological Ramsey number $R^{top}(α,k)$. We prove a topological version of the Erdős-Milner theorem, namely that $R^{top}(α,k)$ is countable whenever $α$ is countable. More precisely, we prove that $R…
We investigate possible cardinalities of maximal antichains in the poset of copies $\langle \mathbb P(\mathbb X),\subset \rangle$ of a countable ultrahomogeneous relational structure $\mathbb X$. It turns out that if the age of $\mathbb X$ has the strong amalgamation property, then, defining a copy of $\mathbb X$ to be large iff it has infinite intersection with each orbit of $\mathbb X$, the structure $\mathbb X$ can be partitioned into countably many large copies, there are almost disjoint families of large copies of size continuum and, hence, there are (maximal) antichains of size continuum…
Stationary common-neighborhood properties and partition hypotheses
本文从大基数出发力迫得到平稳公共邻域性质并改进划分假设的上界，同时证明划分假设在Tukey序下具有传递性。
We study partition properties for uncountable regular cardinals that arise by restricting partition properties defining large cardinal notions to classes of simply definable colourings. We show that both large cardinal assumptions and forcing axioms imply that there is a homogeneous closed unbounded subset of $ω_1$ for every colouring of the finite sets of countable ordinals that is definable by a $Σ_1$-formula that only uses the cardinal $ω_1$ and real numbers as parameters. Moreover, it is shown that certain large cardinal properties cause analogous partition properties to hold at the given …
A sibling of a relational structure $R$ is any structure $S$ which can be embedded into $R$ and, vice versa, in which $R$ can be embedded. Let $sib(R)$ be the number of siblings of $R$, these siblings being counted up to isomorphism. Thomassé conjectured that for countable relational structures made of at most countably many relations, $sib(R)$ is either $1$, countably infinite, or the size of the continuum; but even showing the special case $sib(R)=1$ or infinite is unsettled when $R$ is a countable tree. This is related to Bonato-Tardif conjecture asserting that for every tree $T$ the number…
For a continuous action of a countable discrete group $G$ on a Polish space $X$, a countable Borel partition $P$ of $X$ is called a generator if $G \cdot P := \{ gC : g \in G, C \in P \}$ generates the Borel $σ$-algebra of $X$. For $G = Z$, the Kolmogorov--Sinai theorem gives a measure-theoretic obstruction to the existence of finite generators: they do not exist in the presence of an invariant probability measure with infinite entropy. It was asked by Benjamin Weiss in the late 80s whether the nonexistence of any invariant probability measure guarantees the existence of a finite generator. We…
Stone space partitions indexed by a poset
本文研究由偏序集或PO系统索引的Stone空间修剪划分理论，分析序性质与拓扑性质的关联、划分的扩张完备性、存在充要条件及同胚唯一性，得到Cantor集上满足特定闭包条件的实索引划分推论。
In this issue we celebrate the appearance of the proceedings of the first SPM Workshop, announce several mathematical breakthroughs, have two extended contributions by Babinkostova, and a new open problem by Kalenda. Contents: Editor's note; Proceedings of the first workshop on Coverings, Selections, and Games in Topology; A five element basis for the uncountable linear orders; Set mapping reflection; The Proper Forcing Axiom, Prikry forcing, and the Singular Cardinals Hypothesis; A solution to the L space problem and related ZFC constructions; Countable Tightness, Elementary Submodels and Hom…
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