Paolo Lipparini
We begin the study of the consequences of the existence of certain infinite matrices. Our present application is to compactness of products of topological spaces.
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We generalize the results from "P. Lipparini, Productive $[λ,μ]$-compactness and regular ultrafilters, Topology Proceedings, 21 (1996), 161--171"; in particular the present results apply to singular cardinals, too.
We find many conditions equivalent to the model-theoretical property $λ\stackrelκ{\Rightarrow} μ$ introduced in [L1]. Our conditions involve uniformity of ultrafilters, compactness properties of products of topological spaces and the existence of certain infinite matrices.
We discuss the existence of complete accumulation points of sequences in products of topological spaces. Then we collect and generalize many of the results proved in Parts I, II and IV. The present Part VI is complementary to Part V to the effect that here we deal, say, with uniformity, complete accumulation points and $ κ$-$(λ)$-compactness, rather than with regularity, $[ λ, μ]$-compactness and $ κ$-$ (λ, μ)$-compactness. Of course, if we restrict ourselves to regular cardinals, Parts V (for $ λ= μ$) and Part VI essentially coincide.
We extend to singular cardinals the model-theoretical relation $λ\stackrelκ{\Rightarrow} μ$ introduced in P. Lipparini, The compactness spectrum of abstract logics, large cardinals and combinatorial principles, Boll. Unione Matematica Italiana ser. VII, {\bf 4-B} 875--903 (1990). We extend some results obtained in Part II, finding equivalent conditions involving uniformity of ultrafilters and the existence of certain infinite matrices. Our present definition suggests a new compactness property for abstract logics.
We generalize to the relations $(λ, μ) \stackrelκ{\Rightarrow} (λ', μ')$ and $\alm (λ, μ) \stackrelκ{\Rightarrow} \alm (λ', μ')$ some results obtained in Parts II and IV. We also present a multi-cardinal version.
We define separating properties for normal ultrafilters. We prove that compactness and supercompactness are separable, yet compactness and measurability are not. We describe how to use separating properties in order to elicit distinct normal ultrafilters which do not contain the measurables.
Assuming an abstract comparison principle called the Ultrapower Axiom, which is motivated by the comparison process of inner model theory and generalizes the statement that the Mitchell order is linear on normal ultrafilters, we characterize strongly compact cardinals in terms of supercompactness: a strongly compact cardinal is either supercompact or a limit of supercompact cardinals. Assuming the Ultrapower Axiom and the GCH, we also prove a local result that roughly states that every countably complete ultrafilter factors as a finite iteration of supercompact ultrafilters.
We characterize exactly the compactness properties of the product of κ copies of the space ω with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard elements in elementary estensions. We also have results involving products of possibly uncountable regular cardinals.
Local versions of measurability have been around for a long time. Roughly, one splits the notion of $μ$-completeness into pieces, and asks for a uniform ultrafilter over $μ$ satisfying just some piece of $μ$-completeness. Analogue local versions of weak compactness are harder to come by, since weak compactness cannot be defined by using a single ultrafilter. We deal with the problem by restricting just to a subset $P$ of all the partitions of $μ$ into $<μ$ classes and asking for some ultrafilter $D$ over $μ$ such that no partition in $P$ disproves the $μ$-completeness of $D$. By making $P$ var…
We characterize the compactness properties of the product of λ copies of the space ω with the discrete topology, dealing in particular with the case λ singular, using regular and uniform ultrafilters, infinitary languages and nonstandard elements. We also deal with products of uncountable regular cardinals with the order topology.
We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters. We also list some problems, and furnish applications to topological spaces and to extended logics.
In this paper we analyse and compare two different notions of regularity for filters on complete Boolean algebras. We also announce two results from a forthcoming paper in preparation, which provide a characterization of Keisler's order in terms of Boolean ultrapowers.
We introduce a covering notion depending on two cardinals, which we call $\mathcal O $-$ [ μ, λ]$-compactness, and which encompasses both pseudocompactness and many other generalizations of pseudocompactness. For Tychonoff spaces, pseudocompactness turns out to be equivalent to $\mathcal O $-$ [ ω, ω ]$-compactness. We provide several characterizations of $\mathcal O $-$ [ μ, λ]$-compactness, and we discuss its connection with $D$-pseudocompactness, for $D$ an ultrafilter. We analyze the behaviour of the above notions with respect to products. Finally, we show that our results hold in a more g…
A model for global compactness
本文在超可compact基数一致性的假设下，构造了一个集合论模型，其中每个奇异基数λ上存在一个均匀超滤子，满足对于所有比λ的紧致度大的基数θ，该超滤子是θ-不可分解的，并展示了色数紧致性的实例，推翻Hedetniemi猜想的反例在ZFC中的最优性。
Squares, ultrafilters and forcing axioms
该研究探讨集合论中平方序列、超滤子与力迫公理的关联，证明强力迫公理可与弱带索引平方序列相容、带索引平方序列可与合适参数的不可分解超滤子共存，且正常力迫公理下承载一致不可分解超滤子的基数性质结论最优，推进了集合论紧致性原理的研究。
Contents: 1. Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces, I; 2. Frechet-Urysohn fans in free topological groups; 3. Packing index of subsets in Polish groups; 4. Symmetric monochromatic subsets in colorings of the Lobachevsky plane; 5. Structural Ramsey theory of metric spaces and topological dynamics of isometry groups; 6. Distinguishing Number of Countable Homogeneous Relational Structures; 7. Indestructible colourings and rainbow Ramsey theorems; 8. Products of Borel subgroups; 9. Selection theorems and treeabili…
Contents: 1. Editor's note 2. Research announcements 2.1. Pseudocompact group topologies with no infinite compact subsets 2.2. Selective coideals on (FIN[1]k) 2.3. Entire functions mapping uncountable dense sets of reals onto each other monotonically 2.4. Effective refining of Borel coverings 2.5. Symmetry and colorings: Some results and open problems 2.6. Many partition relations below density 2.7. Lindelof indestructibility, topological games and selection principles 2.8. Locally precompact groups: (Local) realcompactness and connectedness 2.9. The group Aut is Roelcke precompact 2.10. An in…
Recently, S.~Kanti Patra and Md.~Moid Shaik proved the existence of monochromatic solutions to systems of polynomial equations near zero for particular dense subsemigroups $S$ of $((0,\infty),+)$. We extend their results to a much larger class of systems whilst weakening the requests on $S$, using solely basic results about ultrafilters.
This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrapower Axiom, a principle that is expected to hold in Woodin's hypothesized Ultimate \(L\), providing some evidence for the Ultimate \(L\) Conjecture. We show that every regular cardinal above the first strongly compact that carries an indecomposable ultrafilter is measurable, answering a question of Silver for large enough cardinals. We show that any successor almost strongly compact cardinal of uncountable cofinality…
In 2011, Neil Hindman proved that for every natural number $n,m$ the polynomial \begin{equation*} \sum_{i=1}^{n} x_{i}-\prod\limits_{j=1}^{m} y_{j} \end{equation*} has monochromatic solutions for every finite coloration of $\mathbb{N}$. We want to generalize this result to two classes of nonlinear polynomials. The first class consists of polynomials $P(x_{1},...,x_{n},y_{1},...,y_{m})$ of the following kind: \begin{equation*} P(x_{1},...,x_{n},y_{1},...,y_{m})=\sum_{i=1}^{n}a_{i}x_{i}M_{i}(y_{1},...,y_{m}), \end{equation*} where $n,m$ are natural numbers, $\sum\limits_{i=1}^{n}a_{i}x_{i}$ has …
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