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Singular cofinality club_AD Absoluteness incompactness free Souslin tree Uniformly homogeneous Generalized Clubs Strong coloring Subadditive Microscopic Approach Sierpinski's onto mapping principle Foundations nonmeager set Fast club positive partition relation Selective Ultrafilter Monotonically far Generalized descriptive set theory Luzin set SNR b-scale countably metacompact HOD Coherent tree Constructible Universe Reflecting stationary set Fat stationary set ZFC construction Non-saturation Partition Relations Subnormal ideal Parameterized proxy principle Prikry-type forcing Ulam matrix Diamond S-Space Ascending path Cardinal function approachability ideal PFA Hereditarily Lindelöf space Reduced Power Was Ulam right? Partition relations for trees xbox strongly bounded groups Postprocessing function Knaster and friends Small forcing Ascent Path middle diamond Dushnik-Miller Chromatic number Poset Almost Souslin Whitehead Problem polarized partition relation Vanishing levels perfectly normal indecomposable filter Successor of Singular Cardinal GMA PFA(S)[S] Club Guessing Precaliber Strongly Luzin set square principles projective Boolean algebra Greatly Mahlo Minimal Walks Aronszajn tree OCA Chang's conjecture Entangled linear order Forcing with side conditions Commutative projection system Forcing Large Cardinals Iterated forcing Square-Brackets Partition Relations specializable Souslin tree Hedetniemi's conjecture AIM forcing Erdos-Hajnal graphs tensor product graph Ineffable cardinal L-space Mandelbrot set Lipschitz reduction Distributive tree Respecting tree Antichain stick diamond star weak diamond stationary hitting Universal Sequences Fodor-type reflection weak square Knaster Uniformization Axiom R Weakly compact cardinal Cardinal Invariants Well-behaved magma C-sequence Forcing Axioms Prevalent singular cardinals Singular cardinals combinatorics Rainbow sets Almost-disjoint family Slim tree Diamond-sharp coloring number Jonsson cardinal Dowker space Open Access Kurepa Hypothesis free Boolean algebra weak Kurepa tree Ostaszewski square super-Souslin tree unbounded function stationary reflection sap Analytic sets Local Club Condensation. Shelah's Strong Hypothesis Cohen real Singular Density Closed coloring Strongly compact cardinal Amenable C-sequence Interval topology on trees O-space Rock n' Roll Hindman's Theorem Commutative cancellative semigroups Almost countably chromatic Souslin Tree 54G20 Rado's conjecture P-Ideal Dichotomy full tree Uniformly coherent reflection principles Ramsey theory over partitions Successor of Regular Cardinal Intersection model transformations Sigma-Prikry Subtle tree property Subtle cardinal Filter reflection very good scale square Martin's Axiom regressive Souslin tree Countryman line higher Baire space Erdos Cardinal Sakurai's Bell inequality ccc Diamond for trees Nonspecial tree
Tag Archives: Minimal Walks
Was Ulam right? III: Indecomposable ideals
Joint work with Tanmay Inamdar. Abstract. We continue our study of Ulam’s measure problem. In contrast to our previous works, we shift our focus from measures stratified by their additivity, to measures stratified by their indecomposability. The breakthrough here is … Continue reading
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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Complicated colorings, revisited
Joint work with Jing Zhang. Abstract. In a paper from 1997, Shelah asked whether $Pr_1(\lambda^+,\lambda^+,\lambda^+,\lambda)$ holds for every inaccessible cardinal $\lambda$. Here, we prove that an affirmative answer follows from $\square(\lambda^+)$. Furthermore, we establish that for every pair $\chi<\kappa$ of … Continue reading
Strongest transformations
Joint work with Jing Zhang. Abstract. We continue our study of maps transforming high-dimensional complicated objects into squares of stationary sets. Previously, we proved that many such transformations exist in ZFC, and here we address the consistency of the strongest … Continue reading
Posted in Partition Relations, Publications
Tagged Diamond, Minimal Walks, square, Square-Brackets Partition Relations, stick, transformations, xbox
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Transformations of the transfinite plane
Joint work with Jing Zhang. Abstract. We study the existence of transformations of the transfinite plane that allow one to reduce Ramsey-theoretic statements concerning uncountable Abelian groups into classical partition relations for uncountable cardinals. To exemplify: we prove that for every … Continue reading
11th Young Set Theory Workshop, June 2018
I gave a 4-lecture tutorial at the 11th Young Set Theory Workshop, Lausanne, June 2018. Title: In praise of C-sequences. Abstract. Ulam and Solovay showed that any stationary set may be split into two. Is it also the case that … Continue reading
Posted in Invited Talks
Tagged Aronszajn tree, C-sequence, incompactness, Knaster, Minimal Walks, Postprocessing function, square
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6th European Set Theory Conference, July 2017
I gave a 3-lecture tutorial at the 6th European Set Theory Conference in Budapest, July 2017. Title: Strong colorings and their applications. Abstract. Consider the following questions. Is the product of two $\kappa$-cc partial orders again $\kappa$-cc? Does there exist … Continue reading
Posted in Invited Talks, Open Problems
Tagged b-scale, Cohen real, Luzin set, Minimal Walks, Souslin Tree, Square-Brackets Partition Relations
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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
Square with built-in diamond-plus
Joint work with Ralf Schindler. Abstract. We formulate combinatorial principles that combine the square principle with various strong forms of diamond, and prove that the strongest amongst them holds in $L$ for every infinite cardinal. As an application, we prove that … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Almost Souslin, diamond star, Kurepa Hypothesis, Minimal Walks, Respecting tree, square, xbox
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Chain conditions of products, and weakly compact cardinals
Abstract. The history of productivity of the $\kappa$-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every … Continue reading
Posted in Partition Relations, Publications
Tagged Aronszajn tree, ccc, Fat stationary set, Minimal Walks, square, Weakly compact cardinal
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