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tensor product graph Ineffable cardinal Erdos-Hajnal graphs Lipschitz reduction Subnormal ideal Knaster Analytic sets Rado's conjecture ccc Diamond-sharp Amenable C-sequence Square-Brackets Partition Relations stick Generalized Clubs strongly bounded groups Weakly compact cardinal Slim tree Non-saturation polarized partition relation Fat stationary set Uniformly homogeneous Ramsey theory over partitions regressive Souslin tree Chang's conjecture specializable Souslin tree square principles Sigma-Prikry free Boolean algebra indecomposable ultrafilter Uniformly coherent Parameterized proxy principle Subtle cardinal Aronszajn tree Closed coloring full tree Microscopic Approach Whitehead Problem Erdos Cardinal Hindman's Theorem nonmeager set Singular cardinals combinatorics Prevalent singular cardinals diamond star Reduced Power Partition Relations Rock n' Roll Shelah's Strong Hypothesis Iterated forcing Successor of Singular Cardinal Greatly Mahlo Cardinal Invariants Was Ulam right Successor of Regular Cardinal stationary reflection PFA reflection principles Kurepa Hypothesis Coherent tree Hedetniemi's conjecture Strong coloring Ulam matrix Singular cofinality Sierpinski's onto mapping principle Filter reflection middle diamond sap Generalized descriptive set theory free Souslin tree S-Space Forcing Subtle tree property Strongly Luzin set b-scale Cardinal function Souslin Tree L-space Almost-disjoint family Open Access Distributive tree coloring number Forcing Axioms positive partition relation GMA Large Cardinals very good scale Poset transformations Diamond for trees square Axiom R Reflecting stationary set unbounded function Hereditarily Lindelöf space Precaliber Fodor-type reflection Rainbow sets O-space AIM forcing Minimal Walks Constructible Universe Antichain super-Souslin tree Foundations Well-behaved magma PFA(S)[S] Mandelbrot set projective Boolean algebra Dowker space Jonsson cardinal weak diamond incompactness weak square Chromatic number OCA club_AD Uniformization Prikry-type forcing Cohen real Small forcing Sakurai's Bell inequality Absoluteness P-Ideal Dichotomy approachability ideal HOD SNR ZFC construction Fast club Dushnik-Miller Almost countably chromatic Diamond Commutative cancellative semigroups Universal Sequences Almost Souslin countably metacompact Club Guessing Singular Density Knaster and friends Nonspecial tree stationary hitting higher Baire space 54G20 Postprocessing function Luzin set Subadditive Ascent Path xbox Martin's Axiom C-sequence Ostaszewski square Selective Ultrafilter Vanishing levels Local Club Condensation.
Tag Archives: Minimal Walks
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, 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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MFO workshop in Set Theory, January 2014
I gave an invited talk at the Set Theory workshop in Obwerwolfach, January 2014. Talk Title: 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. Downloads: