Archives
Keywords
Dowker space Fast club Precaliber Monotonically far weak diamond Diamond Strongly compact cardinal Was Ulam right? Club Guessing Ulam matrix Aronszajn tree Dushnik-Miller nonmeager set projective Boolean algebra Diamond-sharp Sakurai's Bell inequality Rainbow sets square weak square indecomposable filter Almost countably chromatic Partition relations for trees tensor product graph Rock n' Roll Postprocessing function club_AD GMA AIM forcing Intersection model Constructible Universe Analytic sets Forcing Axioms ccc Uniformization Well-behaved magma Whitehead Problem Ramsey theory over partitions Rado's conjecture Subtle cardinal Subnormal ideal Hindman's Theorem Greatly Mahlo Generalized descriptive set theory Distributive tree O-space Hereditarily Lindelöf space b-scale Commutative projection system Commutative cancellative semigroups diamond star transformations square principles Kurepa Hypothesis Jonsson cardinal free Boolean algebra unbounded function Subtle tree property Hedetniemi's conjecture Reflecting stationary set C-sequence Mandelbrot set Microscopic Approach Poset ZFC construction specializable Souslin tree Almost-disjoint family Generalized Clubs Universal Sequences Singular cardinals combinatorics Sigma-Prikry Filter reflection Axiom R Large Cardinals Lipschitz reduction Nonspecial tree Ascending path Local Club Condensation. Weakly compact cardinal Cardinal Invariants Strong coloring Parameterized proxy principle PFA S-Space middle diamond Prevalent singular cardinals Chang's conjecture Knaster and friends regressive Souslin tree Singular Density Entangled linear order OCA Uniformly homogeneous Singular cofinality PFA(S)[S] Absoluteness coloring number Reduced Power HOD Selective Ultrafilter Sierpinski's onto mapping principle Interval topology on trees very good scale Antichain L-space Forcing with side conditions higher Baire space Successor of Regular Cardinal Ineffable cardinal Open Access Subadditive Amenable C-sequence incompactness Strongly Luzin set Erdos-Hajnal graphs Prikry-type forcing Martin's Axiom Diamond for trees Luzin set free Souslin tree stationary hitting perfectly normal Fat stationary set Square-Brackets Partition Relations countably metacompact stationary reflection approachability ideal polarized partition relation Ascent Path reflection principles Countryman line Uniformly coherent Non-saturation Ostaszewski square Minimal Walks Shelah's Strong Hypothesis Erdos Cardinal Souslin Tree positive partition relation xbox Chromatic number Successor of Singular Cardinal sap weak Kurepa tree full tree Forcing Coherent tree SNR Closed coloring Foundations Slim tree Vanishing levels Small forcing 54G20 Partition Relations Fodor-type reflection strongly bounded groups P-Ideal Dichotomy Almost Souslin Cohen real Iterated forcing Knaster Cardinal function super-Souslin tree Respecting tree stick
Category Archives: Publications
Partitioning a reflecting stationary set
Joint work with Maxwell Levine. Abstract. We address the question of whether a reflecting stationary set may be partitioned into two or more reflecting stationary subsets, providing various affirmative answers in ZFC. As an application to singular cardinals combinatorics, we infer … Continue reading
Souslin trees at successors of regular cardinals
Abstract. We present a weak sufficient condition for the existence of Souslin trees at successor of regular cardinals. The result is optimal and simultaneously improves an old theorem of Gregory and a more recent theorem of the author. Downloads: Citation … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged Parameterized proxy principle, Souslin Tree
1 Comment
Knaster and friends I: Closed colorings and precalibers
Joint work with Chris Lambie-Hanson. Abstract. The productivity of the $\kappa$-chain condition, where $\kappa$ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970s, consistent examples of $\kappa$-cc posets whose squares … Continue reading
A remark on Schimmerling’s question
Joint work with Ari Meir Brodsky. Abstract. Schimmerling asked whether $\square^*_\lambda$ together with GCH entails the existence of a $\lambda^+$-Souslin tree, for a singular cardinal $\lambda$. Here, we provide an affirmative answer under the additional assumption that there exists a … Continue reading
Weak square and stationary reflection
Joint work with Gunter Fuchs. Abstract. It is well-known that the square principle $\square_\lambda$ entails the existence of a non-reflecting stationary subset of $\lambda^+$, whereas the weak square principle $\square^*_\lambda$ does not. Here we show that if $\mu^{cf(\lambda)}<\lambda$ for all $\mu<\lambda$, … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, stationary reflection, weak square
Leave a comment
A forcing axiom deciding the generalized Souslin Hypothesis
Joint work with Chris Lambie-Hanson. Abstract. We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $\lambda$, … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, Souslin Tree, square, super-Souslin tree
1 Comment
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
The eightfold way
Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing … Continue reading
Reflection on the coloring and chromatic numbers
Joint work with Chris Lambie-Hanson. Abstract. We prove that reflection of the coloring number of graphs is consistent with non-reflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large gaps) … Continue reading
Posted in Compactness, Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, Chang's conjecture, Chromatic number, coloring number, Fodor-type reflection, incompactness, Iterated forcing, Parameterized proxy principle, Postprocessing function, Rado's conjecture, square, stationary reflection
2 Comments
Strong failures of higher analogs of Hindman’s Theorem
Joint work with David J. Fernández Bretón. Abstract. We show that various analogs of Hindman’s Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1. There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that … Continue reading
Posted in Groups, Partition Relations, Publications
Tagged 03E02, 03E35, 03E75, 05A17, 05D10, 11P99, 20M14, Chang's conjecture, Commutative cancellative semigroups, Entangled linear order, Erdos Cardinal, Hindman's Theorem, Jonsson cardinal, Kurepa Hypothesis, Square-Brackets Partition Relations, Weakly compact cardinal, ZFC construction
1 Comment
