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Martin's Axiom Cardinal Invariants higher Baire space Foundations Small forcing Rainbow sets Diamond Subtle tree property 54G20 stationary hitting Forcing stationary reflection Intersection model Cohen real indecomposable ultrafilter weak Kurepa tree Knaster Diamond for trees Almost Souslin Sigma-Prikry Respecting tree PFA Commutative projection system Postprocessing function C-sequence Commutative cancellative semigroups countably metacompact free Boolean algebra Erdos Cardinal Erdos-Hajnal graphs Analytic sets Coherent tree Almost countably chromatic Weakly compact cardinal reflection principles Non-saturation Antichain Jonsson cardinal Hedetniemi's conjecture Uniformly homogeneous Fast club Ramsey theory over partitions coloring number Subtle cardinal Generalized descriptive set theory stick Open Access GMA Absoluteness Constructible Universe Strongly Luzin set Singular cofinality SNR unbounded function Rock n' Roll Minimal Walks super-Souslin tree Rado's conjecture Greatly Mahlo sap Prevalent singular cardinals square principles Dushnik-Miller Fodor-type reflection Prikry-type forcing Poset approachability ideal free Souslin tree S-Space P-Ideal Dichotomy Square-Brackets Partition Relations weak square Generalized Clubs club_AD diamond star Subnormal ideal strongly bounded groups Universal Sequences b-scale Filter reflection Sierpinski's onto mapping principle AIM forcing PFA(S)[S] Shelah's Strong Hypothesis Diamond-sharp Ineffable cardinal Kurepa Hypothesis transformations Souslin Tree middle diamond L-space Luzin set Whitehead Problem polarized partition relation Uniformly coherent Almost-disjoint family regressive Souslin tree Successor of Singular Cardinal nonmeager set Vanishing levels Club Guessing Amenable C-sequence Selective Ultrafilter Ascent Path Partition Relations Reduced Power Ostaszewski square Was Ulam right? Hindman's Theorem O-space Chang's conjecture Distributive tree specializable Souslin tree Aronszajn tree Parameterized proxy principle Strong coloring Cardinal function Nonspecial tree weak diamond Successor of Regular Cardinal HOD Ulam matrix very good scale square Forcing Axioms ccc projective Boolean algebra Well-behaved magma Sakurai's Bell inequality Strongly compact cardinal Axiom R Dowker space Closed coloring Mandelbrot set Knaster and friends Microscopic Approach xbox Iterated forcing Chromatic number Countryman line Uniformization Lipschitz reduction Slim tree OCA tensor product graph Fat stationary set Hereditarily Lindelöf space Local Club Condensation. Precaliber Subadditive Reflecting stationary set Large Cardinals full tree positive partition relation ZFC construction Singular Density incompactness Singular cardinals combinatorics
Category Archives: Publications
Knaster and friends III: Subadditive colorings
Joint work with Chris Lambie-Hanson. Abstract. We continue our study of strongly unbounded colorings, this time focusing on subadditive maps. In Part I of this series, we showed that, for many pairs of infinite cardinals $\theta < \kappa$, the existence … 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
2 Comments
On the ideal J[kappa]
Abstract. Motivated by a question from a recent paper by Gilton, Levine and Stejskalova, we obtain a new characterization of the ideal $J[\kappa]$, from which we confirm that $\kappa$-Souslin trees exist in various models of interest. As a corollary we … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged Cardinal Invariants, Cohen real, nonmeager set
1 Comment
Ramsey theory over partitions III: Strongly Luzin sets and partition relations
Joint work with Menachem Kojman and Juris Steprāns. Abstract. The strongest type of coloring of pairs of countable ordinals, gotten by Todorcevic from a strongly Luzin set, is shown to be equivalent to the existence of a nonmeager set of … Continue reading
Ramsey theory over partitions I: Positive Ramsey relations from forcing axioms
Joint work with Menachem Kojman and Juris Steprāns. Abstract. In this series of papers, we advance Ramsey theory of colorings over partitions. In this part, a correspondence between anti-Ramsey properties of partitions and chain conditions of the natural forcing notions … Continue reading
Posted in Partition Relations, Publications
Tagged 03E02, 03E17, 03E35, GMA, Martin's Axiom, positive partition relation, Ramsey theory over partitions
1 Comment
A guessing principle from a Souslin tree, with applications to topology
Joint work with Roy Shalev. Abstract. We introduce a new combinatorial principle which we call $\clubsuit_{AD}$. This principle asserts the existence of a certain multi-ladder system with guessing and almost-disjointness features, and is shown to be sufficient for carrying out … Continue reading
Posted in Publications, Souslin Hypothesis, Topology
Tagged club_AD, Dowker space, O-space, regressive Souslin tree, S-Space, Souslin Tree, Vanishing levels
2 Comments
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
Fake Reflection
Joint work with Gabriel Fernandes and Miguel Moreno. Abstract. We introduce a generalization of stationary set reflection which we call filter reflection, and show it is compatible with the axiom of constructibility as well as with strong forcing axioms. We … Continue reading
A microscopic approach to Souslin-tree constructions. Part II
Joint work with Ari Meir Brodsky. Abstract. In Part I of this series, we presented the microscopic approach to Souslin-tree constructions, and argued that all known $\diamondsuit$-based constructions of Souslin trees with various additional properties may be rendered as applications of … Continue reading
Sigma-Prikry forcing II: Iteration Scheme
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. In Part I of this series, we introduced a class of notions of forcing which we call $\Sigma$-Prikry, and showed that many of the known Prikry-type notions of forcing that centers … Continue reading
