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