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specializable Souslin tree full tree Cohen real PFA xbox Almost Souslin transformations Constructible Universe Was Ulam right? Diamond for trees Almost-disjoint family Dushnik-Miller HOD Ascent Path stationary reflection Absoluteness Mandelbrot set Rainbow sets S-Space OCA Subtle cardinal PFA(S)[S] ZFC construction Weakly compact cardinal Small forcing L-space strongly bounded groups Fat stationary set Rock n' Roll Slim tree P-Ideal Dichotomy reflection principles Amenable C-sequence Fast club Square-Brackets Partition Relations Lipschitz reduction Uniformization Almost countably chromatic Well-behaved magma Ramsey theory over partitions Respecting tree Axiom R Singular cofinality Diamond-sharp diamond star GMA Hereditarily Lindelöf space Countryman line stick club_AD Uniformly coherent approachability ideal Kurepa Hypothesis nonmeager set Whitehead Problem Commutative projection system Large Cardinals weak diamond Shelah's Strong Hypothesis Precaliber Club Guessing Knaster and friends Souslin Tree Chromatic number Closed coloring polarized partition relation Poset Partition Relations Postprocessing function Successor of Regular Cardinal square principles Forcing unbounded function coloring number very good scale Uniformly homogeneous Microscopic Approach super-Souslin tree Local Club Condensation. Generalized descriptive set theory Ostaszewski square projective Boolean algebra Hindman's Theorem Subtle tree property Sierpinski's onto mapping principle Selective Ultrafilter weak square middle diamond Prikry-type forcing tensor product graph Martin's Axiom Jonsson cardinal incompactness Generalized Clubs Fodor-type reflection Knaster regressive Souslin tree Filter reflection Cardinal function positive partition relation Minimal Walks C-sequence Chang's conjecture Forcing Axioms higher Baire space Greatly Mahlo Reflecting stationary set sap Open Access Sakurai's Bell inequality Erdos Cardinal countably metacompact Dowker space Strong coloring Ulam matrix Non-saturation ccc Distributive tree Prevalent singular cardinals indecomposable ultrafilter free Boolean algebra SNR Luzin set Foundations b-scale Analytic sets square Successor of Singular Cardinal Hedetniemi's conjecture Universal Sequences Intersection model Antichain Singular Density Commutative cancellative semigroups Reduced Power Sigma-Prikry Iterated forcing Coherent tree Nonspecial tree Rado's conjecture Singular cardinals combinatorics O-space Erdos-Hajnal graphs Strongly Luzin set weak Kurepa tree Cardinal Invariants Subadditive Ineffable cardinal Diamond Aronszajn tree Strongly compact cardinal stationary hitting 54G20 Parameterized proxy principle Subnormal ideal Vanishing levels AIM forcing free Souslin tree
Category Archives: Squares and Diamonds
Diamond on Kurepa trees
Joint work with Ziemek Kostana and Saharon Shelah. Abstract. We introduce a new weak variation of diamond that is meant to only guess the branches of a Kurepa tree. We demonstrate that this variation is considerably weaker than diamond by … Continue reading
Posted in Preprints, Squares and Diamonds
Tagged Diamond, Diamond for trees, Iterated forcing, Kurepa Hypothesis, weak Kurepa tree
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A club guessing toolbox I
Joint work with Tanmay Inamdar. Abstract. Club guessing principles were introduced by Shelah as a weakening of Jensen’s diamond. Most spectacularly, they were used to prove Shelah’s ZFC bound on the power of the first singular cardinal. These principles have … Continue reading
A new small Dowker space
Joint work with Roy Shalev and Stevo Todorcevic. Abstract. It is proved that if there exists a Luzin set, or if either the stick principle or
Inclusion modulo nonstationary
Joint work with Gabriel Fernandes and Miguel Moreno. Abstract. A classical theorem of Hechler asserts that the structure
Weak square and stationary reflection
Joint work with Gunter Fuchs. Abstract. It is well-known that the square principle
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, stationary reflection, weak square
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Distributive Aronszajn trees
Joint work with Ari Meir Brodsky. Abstract. Ben-David and Shelah proved that if
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
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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Putting a diamond inside the square
Abstract. By a 35-year-old theorem of Shelah,
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Diamond, square, Successor of Singular Cardinal
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The search for diamonds
Abstract: This is a review I wrote for the Bulletin of Symbolic Logic on the following papers: Saharon Shelah, Middle Diamond, Archive for Mathematical Logic, vol. 44 (2005), pp. 527–560. Saharon Shelah, Diamonds, Proceedings of the American Mathematical Society, vol. … Continue reading
Posted in Publications, Reviews, Squares and Diamonds
Tagged Diamond, middle diamond, weak diamond, weak square
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Jensen’s diamond principle and its relatives
This is chapter 6 in the book Set Theory and Its Applications (ISBN: 0821848127). Abstract: We survey some recent results on the validity of Jensen’s diamond principle at successor cardinals. We also discuss weakening of this principle such as club … Continue reading
