Archives
Keywords
stationary hitting Dowker space Monotonically far Almost-disjoint family Coherent tree Intersection model Singular cofinality Hedetniemi's conjecture Fast club strongly bounded groups nonmeager set b-scale weak Kurepa tree Fodor-type reflection OCA Subadditive Erdos-Hajnal graphs Chang's conjecture Luzin set Fat stationary set Martin's Axiom Rado's conjecture Axiom R projective Boolean algebra Slim tree Weakly compact cardinal Subnormal ideal HOD PFA specializable Souslin tree Constructible Universe Knaster and friends Nonspecial tree Subtle tree property Precaliber Parameterized proxy principle Strongly compact cardinal countably metacompact Rainbow sets Cohen real Uniformization Universal Sequences Absoluteness Uniformly coherent Singular Density Reduced Power coloring number Selective Ultrafilter approachability ideal transformations Microscopic Approach Rock n' Roll Ineffable cardinal Prevalent singular cardinals ZFC construction L-space Greatly Mahlo O-space Successor of Singular Cardinal ccc polarized partition relation Ostaszewski square Partition Relations Commutative projection system Respecting tree Antichain Strongly Luzin set Iterated forcing xbox Dushnik-Miller Forcing Diamond for trees Hindman's Theorem Sakurai's Bell inequality Ulam matrix reflection principles Souslin Tree GMA Strong coloring club_AD positive partition relation incompactness sap free Souslin tree diamond star P-Ideal Dichotomy Ascent Path Analytic sets Mandelbrot set Small forcing Whitehead Problem Non-saturation square principles Distributive tree Interval topology on trees Sigma-Prikry stationary reflection Postprocessing function higher Baire space Uniformly homogeneous Large Cardinals Ascending path indecomposable filter Was Ulam right? Generalized descriptive set theory Commutative cancellative semigroups Almost countably chromatic square Chromatic number Ramsey theory over partitions Shelah's Strong Hypothesis Club Guessing SNR Jonsson cardinal Countryman line Singular cardinals combinatorics very good scale Prikry-type forcing full tree weak diamond Kurepa Hypothesis Filter reflection C-sequence Aronszajn tree Almost Souslin tensor product graph AIM forcing Entangled linear order perfectly normal Forcing Axioms Square-Brackets Partition Relations S-Space PFA(S)[S] Sierpinski's onto mapping principle Vanishing levels Poset Open Access Local Club Condensation. Amenable C-sequence Foundations Well-behaved magma Hereditarily Lindelöf space Subtle cardinal Partition relations for trees Lipschitz reduction Cardinal Invariants regressive Souslin tree Reflecting stationary set super-Souslin tree Knaster Minimal Walks weak square Generalized Clubs 54G20 Diamond-sharp unbounded function free Boolean algebra stick middle diamond Diamond Erdos Cardinal Closed coloring Cardinal function Forcing with side conditions Successor of Regular Cardinal
Tag Archives: Club Guessing
Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines
Joint work with Tanmay Inamdar. Abstract. We investigate global structural properties of linear orders of a fixed infinite size. It is classical that the countable linear orders and the continuum-sized orders exhibit contrasting behaviours. Modern results show that strong extensions … Continue reading
Posted in Basis problems, Partition Relations, Preprints
Tagged Aronszajn tree, Ascending path, Club Guessing, Countryman line, Entangled linear order, Minimal Walks, Monotonically far, Partition relations for trees, Strong coloring, Subtle tree property, Vanishing levels, ZFC construction
2 Comments
Winter School in Abstract Analysis, January 2023
I gave a 3-lecture tutorial at the Winter School in Abstract Analysis in Steken, January 2023. Title: Club guessing Abstract. Club guessing principles were introduced by Shelah as a weakening of Jensen’s diamond. Most spectacularly, they were used to prove … Continue reading
A club guessing toolbox I
Joint work with Tanmay Inamdar. Abstract. Club guessing principles were introduced by Shelah as a weakening of Jensen’s diamond. Most spectacularly, they were used to prove Shelah’s ZFC bound on the power of the first singular cardinal. These principles have … Continue reading
Partitioning a reflecting stationary set
Joint work with Maxwell Levine. Abstract. We address the question of whether a reflecting stationary set may be partitioned into two or more reflecting stationary subsets, providing various affirmative answers in ZFC. As an application to singular cardinals combinatorics, we infer … Continue reading
4th Arctic Set Theory Workshop, January 2019
I gave an invited talk at the Arctic Set Theory Workshop 4 in Kilpisjärvi, January 2019. Talk Title: Splitting a stationary set: Is there another way? Abstract: Motivated by a problem in pcf theory, we seek for a new way … Continue reading
Posted in Invited Talks
Tagged Club Guessing, Reflecting stationary set, Ulam matrix, very good scale
Comments Off on 4th Arctic Set Theory Workshop, January 2019
Distributive Aronszajn trees
Joint work with Ari Meir Brodsky. Abstract. Ben-David and Shelah proved that if $\lambda$ is a singular strong-limit cardinal and $2^\lambda=\lambda^+$, then $\square^*_\lambda$ entails the existence of a $\lambda$-distributive $\lambda^+$-Aronszajn tree. Here, it is proved that the same conclusion remains … Continue reading
Partitioning the club guessing
In a recent paper, I am making use of the following fact. Theorem (Shelah, 1997). Suppose that $\kappa$ is an accessible cardinal (i.e., there exists a cardinal $\theta<\kappa$ such that $2^\theta\ge\kappa)$. Then there exists a sequence $\langle g_\delta:C_\delta\rightarrow\omega\mid \delta\in E^{\kappa^+}_\kappa\rangle$ … Continue reading
Shelah’s approachability ideal (part 2)
In a previous post, we defined Shelah’s approachability ideal $I[\lambda]$. We remind the reader that a subset $S\subseteq\lambda$ is in $I[\lambda]$ iff there exists a collection $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq\mathcal [\mathcal P(\lambda)]^{<\lambda}$ such that for club many $\delta\in S$, the union … Continue reading
Posted in Blog, Expository, Open Problems
Tagged approachability ideal, Club Guessing
Leave a comment
Shelah’s approachability ideal (part 1)
Given an infinite cardinal $\lambda$, Shelah defines an ideal $I[\lambda]$ as follows. Definition (Shelah, implicit in here). A set $S$ is in $I[\lambda]$ iff $S\subseteq\lambda$ and there exists a collection $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq\mathcal [\mathcal P(\lambda)]^{<\lambda}$, and some club $E\subseteq\lambda$, so … Continue reading
An inconsistent form of club guessing
In this post, we shall present an answer (due to P. Larson) to a question by A. Primavesi concerning a certain strong form of club guessing. We commence with recalling Shelah’s concept of club guessing. Concept (Shelah). Given a regular … Continue reading
