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