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Whitehead Problem Was Ulam right? Axiom R Vanishing levels Aronszajn tree L-space Mandelbrot set PFA Chromatic number P-Ideal Dichotomy stationary hitting b-scale Cardinal Invariants Luzin set ccc polarized partition relation Countryman line club_AD Prikry-type forcing O-space Amenable C-sequence Almost Souslin xbox projective Boolean algebra full tree stick Ulam matrix Ascent Path diamond star square Diamond for trees Rado's conjecture sap stationary reflection Analytic sets Weakly compact cardinal Antichain Strongly Luzin set Singular cofinality Erdos-Hajnal graphs Entangled linear order Almost countably chromatic Respecting tree square principles Commutative projection system Forcing Axioms Dushnik-Miller Lipschitz reduction Souslin Tree approachability ideal Fat stationary set Partition relations for trees Generalized Clubs weak Kurepa tree Closed coloring GMA Precaliber countably metacompact Martin's Axiom Universal Sequences weak square Sigma-Prikry Uniformization indecomposable filter Absoluteness Strongly compact cardinal reflection principles Filter reflection Knaster and friends higher Baire space Fodor-type reflection free Souslin tree Cardinal function Rock n' Roll Distributive tree middle diamond Knaster Successor of Singular Cardinal SNR Erdos Cardinal Small forcing Subadditive Subtle tree property Parameterized proxy principle Subnormal ideal Successor of Regular Cardinal unbounded function Sakurai's Bell inequality incompactness Hereditarily Lindelöf space positive partition relation Poset Partition Relations Constructible Universe Hindman's Theorem PFA(S)[S] transformations super-Souslin tree Ineffable cardinal Prevalent singular cardinals C-sequence Jonsson cardinal Singular Density Slim tree coloring number OCA Square-Brackets Partition Relations Ramsey theory over partitions Singular cardinals combinatorics Well-behaved magma Reflecting stationary set AIM forcing regressive Souslin tree Interval topology on trees Kurepa Hypothesis Hedetniemi's conjecture Uniformly coherent Commutative cancellative semigroups Microscopic Approach Sierpinski's onto mapping principle nonmeager set HOD free Boolean algebra Cohen real weak diamond Uniformly homogeneous Fast club perfectly normal Postprocessing function Ostaszewski square ZFC construction Forcing Club Guessing Strong coloring Forcing with side conditions Almost-disjoint family Open Access Foundations 54G20 Local Club Condensation. Large Cardinals Iterated forcing Dowker space Subtle cardinal very good scale Greatly Mahlo Nonspecial tree Ascending path specializable Souslin tree Diamond Intersection model Generalized descriptive set theory Minimal Walks strongly bounded groups Monotonically far Selective Ultrafilter S-Space Chang's conjecture tensor product graph Diamond-sharp Reduced Power Non-saturation Rainbow sets Coherent tree Shelah's Strong Hypothesis
Category Archives: Publications
Was Ulam right? II: Small width and general ideals
Joint work with Tanmay Inamdar. Abstract. We continue our study of Sierpinski-type colourings. In contrast to the prequel paper, we focus here on colourings for ideals stratified by their completeness degree. In particular, improving upon Ulam’s theorem and its extension … Continue reading
Posted in Partition Relations, Publications
Tagged 03E02, 03E35, 03E55, C-sequence, Kurepa Hypothesis, Open Access, Subnormal ideal, Ulam matrix, Was Ulam right?
2 Comments
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
Sigma-Prikry forcing III: Down to Aleph_omega
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. We prove the consistency of the failure of the singular cardinals hypothesis at $\aleph_\omega$ together with the reflection of all stationary subsets of $\aleph_{\omega+1}$. This shows that two classical results of … Continue reading
Was Ulam right? I: Basic theory and subnormal ideals
Joint work with Tanmay Inamdar. Abstract. We introduce various coloring principles which generalize the so-called onto mapping principle of Sierpinski to larger cardinals and general ideals. We prove that these principles capture the notion of an Ulam matrix and allow … Continue reading
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
