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strongly bounded groups Hedetniemi's conjecture weak Kurepa tree Absoluteness free Souslin tree PFA Subtle tree property Whitehead Problem Luzin set Rock n' Roll Was Ulam right? positive partition relation diamond star Dowker space Ulam matrix Ineffable cardinal Subnormal ideal Prikry-type forcing Generalized Clubs Ascending path Hereditarily Lindelöf space Prevalent singular cardinals unbounded function Subadditive Successor of Regular Cardinal super-Souslin tree Selective Ultrafilter Martin's Axiom Souslin Tree Sakurai's Bell inequality Hindman's Theorem sap Aronszajn tree Diamond for trees square Small forcing Greatly Mahlo Sigma-Prikry Fodor-type reflection indecomposable filter Ascent Path ccc coloring number Singular cofinality Closed coloring full tree SNR Axiom R HOD L-space Strongly Luzin set Successor of Singular Cardinal Foundations Ostaszewski square Forcing with side conditions Diamond square principles Generalized descriptive set theory Cardinal Invariants stationary hitting Fat stationary set Rado's conjecture Weakly compact cardinal Minimal Walks b-scale Constructible Universe Almost Souslin P-Ideal Dichotomy Partition Relations Respecting tree incompactness Entangled linear order Knaster Distributive tree Mandelbrot set Singular Density stationary reflection weak square Ramsey theory over partitions Square-Brackets Partition Relations Non-saturation Commutative cancellative semigroups Forcing Axioms xbox Singular cardinals combinatorics Analytic sets C-sequence specializable Souslin tree nonmeager set free Boolean algebra Chang's conjecture Reduced Power Iterated forcing Jonsson cardinal Forcing Strongly compact cardinal Antichain Interval topology on trees polarized partition relation higher Baire space tensor product graph Shelah's Strong Hypothesis Monotonically far Fast club Cardinal function O-space AIM forcing Strong coloring Amenable C-sequence Diamond-sharp PFA(S)[S] Filter reflection Almost-disjoint family Sierpinski's onto mapping principle Universal Sequences GMA Knaster and friends Intersection model Vanishing levels reflection principles club_AD approachability ideal Slim tree countably metacompact Subtle cardinal Partition relations for trees transformations middle diamond Countryman line very good scale Poset Reflecting stationary set Open Access stick Uniformization Commutative projection system Almost countably chromatic weak diamond Cohen real Uniformly homogeneous Large Cardinals OCA Well-behaved magma Lipschitz reduction Erdos Cardinal perfectly normal Microscopic Approach Uniformly coherent Dushnik-Miller Erdos-Hajnal graphs Chromatic number ZFC construction S-Space Nonspecial tree projective Boolean algebra Coherent tree Club Guessing Local Club Condensation. Rainbow sets Postprocessing function Precaliber regressive Souslin tree Parameterized proxy principle Kurepa Hypothesis 54G20
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
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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
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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
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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
