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weak Kurepa tree Almost Souslin square principles P-Ideal Dichotomy Large Cardinals Rainbow sets ccc Ramsey theory over partitions Successor of Singular Cardinal Whitehead Problem ZFC construction Hindman's Theorem positive partition relation Chang's conjecture O-space Subnormal ideal Precaliber C-sequence SNR Absoluteness Shelah's Strong Hypothesis Sakurai's Bell inequality Sierpinski's onto mapping principle Strong coloring Erdos Cardinal L-space Martin's Axiom unbounded function Jonsson cardinal Respecting tree Fat stationary set very good scale Chromatic number Vanishing levels Slim tree Uniformly coherent Weakly compact cardinal GMA Greatly Mahlo Rock n' Roll Postprocessing function AIM forcing Selective Ultrafilter Square-Brackets Partition Relations reflection principles Mandelbrot set weak square Reduced Power Uniformly homogeneous tensor product graph Singular cofinality xbox Was Ulam right full tree Poset Hedetniemi's conjecture coloring number Club Guessing Iterated forcing Countryman line super-Souslin tree projective Boolean algebra Rado's conjecture Luzin set Prikry-type forcing Forcing Axioms Erdos-Hajnal graphs weak diamond Strongly Luzin set club_AD Antichain Souslin Tree Ostaszewski square Almost-disjoint family S-Space Fodor-type reflection Aronszajn tree polarized partition relation approachability ideal Small forcing Universal Sequences Filter reflection Hereditarily Lindelöf space Coherent tree Uniformization Dushnik-Miller Commutative cancellative semigroups strongly bounded groups Prevalent singular cardinals Nonspecial tree stationary hitting Reflecting stationary set Ulam matrix Microscopic Approach Partition Relations Ascent Path Cardinal Invariants free Souslin tree diamond star Closed coloring Foundations Diamond Cohen real PFA(S)[S] incompactness Constructible Universe Intersection model Minimal Walks Ineffable cardinal Strongly compact cardinal Non-saturation Diamond for trees Singular cardinals combinatorics PFA stick Parameterized proxy principle Subadditive Local Club Condensation. sap Successor of Regular Cardinal Singular Density indecomposable ultrafilter countably metacompact Lipschitz reduction Forcing Commutative projection system middle diamond Almost countably chromatic Dowker space Subtle tree property Cardinal function OCA Knaster Diamond-sharp Generalized Clubs Open Access 54G20 Knaster and friends b-scale HOD square regressive Souslin tree Axiom R Kurepa Hypothesis stationary reflection Amenable C-sequence Analytic sets Subtle cardinal Well-behaved magma higher Baire space Distributive tree Fast club Sigma-Prikry Generalized descriptive set theory transformations specializable Souslin tree nonmeager set free Boolean algebra
Tag Archives: Minimal Walks
Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines
Joint work with Tanmay Inamdar. Abstract. It is proved that if there is an $\aleph_2$-Aronszajn line, then there is one that does not contain an $\aleph_2$-Countryman line. This solves a problem of Moore and stands in a sharp contrast with … Continue reading
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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Complicated colorings
Abstract. If $\lambda,\kappa$ are regular cardinals, $\lambda>\kappa^+$, and $E^\lambda_{\ge\kappa}$ admits a nonreflecting stationary set, then $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ holds. (Recall that $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ asserts the existence of a coloring $d:[\lambda]^2\rightarrow\lambda$ such that for any family $\mathcal A\subseteq[\lambda]^{<\kappa}$ of size $\lambda$, consisting of pairwise … Continue reading
Posted in Partition Relations, Publications
Tagged Minimal Walks, Open Access, Square-Brackets Partition Relations
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