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